-NRLF 


A  TEXT-BOOK 

ON 

ADVANCED  ALGEBRA 

AND 

TRIGONOMETRY 

WITH  TABLES 


BY 


WILLIAM    CHARLES   BRENKE,    Ph.D. 

ASSOCIATE    PROFESSOR    OF   MATHEMATICS    IN 
THE   UNIVERSITY   OF   NEBRASKA 


/^ 


NEW  YORK 
THE    CENTURY   CO. 

1910 


Copyright,  1910,  by 
THE  CENTURY  COMPANY 

Published,  August,  1910 


Stanbopc  iprcaa 

F.    H.   GILSON     COMPANY 

BOSTON,     U.S.A. 


Q 


.^■'-v 


TABLE  OF  CONTENTS 


CHAPTER  I. 

Page 

The  Operations  of  Algebra 3 

(The  numbers  refer  to  articles.) 

1.  Letters  as  Symbols  of  Quantity.  2.  Signs  of  Relation.  3. 
The  Four  Fundamental  Operations.  4.  Rational  Numbers.  5.  Zero.  7. 
Infinity.  8.  Powers.  9.  Some  Important  Relations.  10.  Exercises.  11. 
Factoring.  Factor  Theorem.  12.  Exercises.  13.  Highest  Common 
Factor.  14.  Least  Common  Multiple.  15.  Exercises.  16.  Fractions. 
19.   Exercises. 

CHAPTER   II. 

Involution;  Evolution;  Theory   of   Exponents;    Surds  and  Im- 

AGINARIES 17 

20.  Involution.  Negative  Exponent.  21.  Exercises.  Zero  Exponent. 
22.  Evolution.  23.  Rational  Exponent.  24.  Irrational  Numbers.  25. 
Irrational  Exponents.  26.  Imaginary  Numbers.  27.  Reduction  of 
Surds.     33.   Exercises. 


CHAPTER   III. 
Logarithms;  Binomial  Theorem  for  Positive  Integral  Exponents.       28 

34.  Logarithms.  39.  Laws  of  Operation  with  Logarithms.  41.  Ex- 
ercises. 42.  Binomial  Theorem  for  Positive  Integral  Exponents.  45. 
Exercises.     46.   Approximate  Computation. 

CHAPTER   IV. 

Linear  Equations 37 

48.  Linear  Equation.  50.  Infinite  Solutions.  51.  Exercises.  52. 
Graphic  Solution.  55.  Exercises.  56.  Coordinates.  58.  Use  of  the 
Graph.  59.  Exercises.  60.  Problems.  61.  Simultaneous  Linear 
Equations.  63.  Graphic  Solution.  64.  Exercises.  65.  Three  Equa- 
tions in  Three  Unknowns.  68.  Four  Equations  in  Four  Unknowns. 
69.    Exercises  and  Problems. 

iii 


iv  -  TABLE  OF  CONTENTS 

CHAPTER  V. 

Page 

Quadratic  Equations 54 

72.  Solution  by  Factoring.  73.  Solution  by  Completing  the  Square. 
74.  Solution  by  Formula.  75.  Exercises.  76.  Nature  of  Roots.  Dis- 
criminant. 77.  Exercises.  78.  Relations  between  Coefficients  and 
Roots.  80.  Exercises.  81.  Graphic  Solution.  82.  Parabola.  84.  Ex- 
ercises. 85.  Equations  Reducible  to  Quadratics.  86.  Exercises  and 
Problems.  87.  Simultaneous  Quadratics.  89.  Nature  of  the  Solutions. 
91.  Graphic  Solution.  93.  Standard  Equation  of  the  Circle.  Exercises. 
95.  Standard  Equation  of  the  Ellipse.  Exercises.  97.  Standard  Equa- 
tions of  the  Parabola.  Exercises.  99.  Standard  Equation  of  the 
Hyperbola.  Exercises.  100.  Rectangular  Hyperbola.  102.  Exercises. 
103.  Solution  of  Two  Simultaneous  Quadratics.  11.  Summary  of 
Methods  for  Solving  Simultaneous  Equations.  112.  Exercises.  113. 
Exponential  Equations.     114.   Exercises. 

CHAPTER  VI. 

Ratio,  Proportion,  Variation 88 

115.  Definitions.  116.  Laws  of  Proportion.  Exercises.  118. 
Variation.  119.  Direct  Variation.  120.  Inverse  Variation.  121.  Joint 
Variation.     122.   Exercises. 

CHAPTER  VII. 

The  Trigonometric  Functions 94 

124.  Trigonometric  Functions.  Exercises.  128.  Functions  of  Com- 
plementary Angles.  Cofunctions.  129.  Application  of  the  Trigono- 
metric Functions  to  the  Solution  of  Right  Triangles.  130.  Exercises. 
131.  Angles  of  any  Magnitude.  132.  The  Trigonometric  Functions  of 
any  Angle.  134.  Line  Values.  135.  Variation  of  the  Trigonometric 
Functions.  Graphs  of  the  Trigonometric  Functions.  Exercises.  136. 
Periodicity  of  the  Trigonometric  Functions.  137.  Relations  between 
the  Functions.  138.  Exercises.  142.  Versed  Sine  and  Coversed  Sine. 
Exercises.  143.  Radian  Measure.  144.  Radians  into  degrees,  and 
conversely.  145.  Exercises.  146.  Angles  corresponding  to  a  Given 
Function.  147.  Use  of  Tables  of  Natural  Functions.  148.  Exercises. 
149.   Given  one  function,  to  find  the  other  functions.     150.   Exercises. 

CHAPTER  VIII. 

Functions  of  Several  Angles 121 

1.52.  Functions  of  (x  ±  2/).  Exercises.  156.  Functions  of  2  x.  Exer- 
cises. 157.  Functions  of  ^x.  Exercises.  158.  Addition  Theorems. 
Exercises.     159.   Exercises. 


Paob 


TABLE  OF  CONTENTS 

CHAPTER  IX. 

^  sin  x         tan  .x       ^  ^  „ 

Ratios  — —  and  — = —      Inverse      Functions.      Trigonometric 

A'  -* 

Equations 133 

160.    Ratios ^  and  .     Exercises.     161.   Inverse  Functions, 

X  X 

Exercises.     164.   Trigonometric    Equations.     Graphic    Solution.     Ex- 
ercises. 

CHAPTER  X. 

Oblique  Plane  Triangles 144 

169.  The  Law  of  Sines.  170.  The  Law  of  Cosines.  171.  The  Law 
of  Tangents.  172.  Functions  of  the  Half  Angles.  173.  Solution  of 
Plane  Oblique  Triangles.     Exercises.     179.   Exercises  and  Problems. 

CHAPTER  XL 

The  Progressions,  Interest  and  Annuities 161 

180.  Arithmetic  Progressions.  Exercises.  184.  Geometric  Pro- 
gressions. Exercises.  188.  Infinite  Geometric  Progressions.  Exer- 
cises. 190.  Harmonic  Progressions.  191.  Exercises.  192.  Interest. 
193.   Annuities.     194.    Exercises. 

CHAPTER  XII. 

Infinite  Series 171 

195.  Limit  of  a  Variable  Quantity.  196.  Infinite  Series.  199.  Alter- 
nating Series.  200.  Absolute  Convergence.  201.  The  Comparison 
Test.     202.   The  Ratio  Test.     203.   Exercises. 

CHAPTER  XIII. 

Functions,  Derivatives,  Maclaurin's  Series 179 

204.  Functions.  205.  Variation  of  Functions.  Exercises.  206. 
Difference  Quotient.  208.  Limit  of  D.  Q.  =  Slope  of  Tangent.  209. 
Examples.  Exercises.  210.  Derivative.  211.  Calculation  of  Deriva- 
tives. Exercises.  215.  The  Derivative  as  a  Rate  of  Change.  Exercises. 
217.  Higher  Derivatives.  218.  Maclaurin's  Series.  220.  The  Bino- 
mial Theorem.     222.   Exercises. 

CHAPTER  XIV. 
Computation,  Approximations,  Differences  and  Interpolation.  . .     199 

223.  Remarks  on  Computation.  224.  Useful  Approximations.  Ex- 
ercises. 225.  Computation  of  Logarithms.  Exercises.  227.  Differ- 
ences.    Exercises.     230.   Interpolation.     Exercises. 


vi  TABLE  OF  CONTENTS 

CHAPTER  XV. 

Page 
Undetermined  Coefficients.     Partial  Fractions 210 

234.  Theorem  of  Undetermined  Coefficients.  Exercises.  235.  Par- 
tial Fractions.     239.    Exercises. 

CHAPTER   XVI. 

Determinants 217 

240.  Determinants  of  the  Second  Order.  241.  Determinants  of  the 
Third  Order.  Exercises.  243.  General  Definition  of  a  Determinant. 
247.  Properties  of  Determinants.  248.  Solution  of  Systems  of  Linear 
Equations.     249.   Exercises. 

CHAPTER  XVIL 

Polar  Coordinates.     Complex  Numbers.     De  Moivre's  Theorem. 

Exponential  Values  OF  sin  X  and  cos  X.   Hyberbolic  Functions.     231 

250.  Polar  Coordinates.  252.  Curves  in  Polar  Coordinates.  Exer- 
cises. 253.  Complex  Numbers.  256.  De  Moivre's  Theorem.  259. 
The  nth  Roots  of  Unity.  Exercises.  260.  Expansion  of  sin  ?i0  and  cos  nd. 
Exercises.     261.   Exponential  Values  of  sin  x  and  cos  x.     Exercises. 

CHAPTER  XVIII. 

Permutations.    Combinations.     Chance 242 

263.  Permutations.  Exercises.  264.  Combinations.  Exercises. 
266.  Exercises.  267.  Probability  or  Chance.  Exercises.  270.  Exer- 
cises. 

CHAPTER  XIX. 
Theory  of  Equations 249 

272.  Factor  Theorem.  273.  Depressed  Equation.  Exercises.  274. 
Number  of  Roots.  Exercises.  275.  To  Form  an  Equation  having 
Given  Roots.  Exercises.  276.  Relations  between  Coefficients  and 
Roots.  277.  Fractional  Roots.  278.  Imaginary  Roots.  279.  Multi- 
ple Roots.  280.  Exercises.  281.  Transformation  of  Equations.  282. 
Synthetic  Division.  285.  Occurrence  of  Imaginary  Roots  in  Pairs. 
286.  Exercises.  287.  Approximation  to  the  Roots  of  an  Equation. 
289.  Exercises.  290.  Cardan's  Solution  of  the  Cubic  Equation.  Nature 
of  the  Roots.  291.  Ferrari's  Solution  of  the  Quartic  Equation.  Exer- 
cises. 


TABLE  OF  CONTENTS  vii 

CHAPTER  XX. 

Paob 

Spherical  Trigonometry 2G9 

292.  Spherical  Geometry.  293.  Spherical  Right  Triangles.  294. 
Napier's  Rules  of  Circular  Parts.  297.  Oblique  Triangles.  Law  of 
Sines.  Law  of  Cosines.  298.  Principle  of  Duality.  299.  Formulas 
for  the  Half  Angle.  300.  Formulas  for  the  Half  Sides.  30L  Napier's 
Analogies.  302.  Area  of  a  Spherical  Triangle.  303.  Solution  of  Spheri- 
cal Oblique  Triangles.  305.  Exercises.  306.  Applications  to  the  Ter- 
restrial Sphere.  Exercises.  307.  Applications  to  the  Celestial  Sphere. 
Exercises. 

Answers  to  Odd-Numbered  Exercises 284 

Index 297 

Appendix  A.   List  of  Formulas 301 

Appendix  B.   Tables  I  to  VH 315 

Protractor Inside  of  back  cover     y 


PREFACE 


In  a  considerable  number  of  our  colleges  and  universities  the 
work  of  the  first  semester  in  mathematics  is  devoted  to  Algebra 
and  Trigonometry.  Usually  Algebra  is  taken  up  first  and  then 
Trigonometry,  or  else  the  two  subjects  are  studied  on  alternate 
days.  Neither  plan  is  quite  satisfactory.  It  has  therefore  seemed 
to  the  writer  that  a  single  book,  treating  both  subjects  in  a  corre- 
lated manner,  might  be  of  service  both  to  student  and  teacher. 

In  the  present  text  the  principal  departures  from  the  subject 
matter  usually  treated  will  be  found  in  chapters  13  and  14.  The 
chief  aim  has  been  to  follow  a  mode  and  sequence  of  presentation 
which  shall  introduce  the  student  who  needs  to  apply  his  knowl- 
edge of  mathematics  in  his  other  work  as  directly  as  possible  to 
those  facts  and  concepts  which  are  most  useful  to  him. 

For  this  reason  much  stress  is  laid  on  graphic  methods  in  the 
chapters  on  linear  and  quadratic  equations,  and  this  is  followed 
up  later  as  opportunity  arises.  It  is  thought  that  the  extra  time 
so  used  will  be  more  than  made  up  when  the  student  begins  his 
study  of  Analytical  Geometry,  because  he  will  have  become  grad- 
ually familiar  with  the  fundamental  idea  of  this  subject  and  need 
not  readjust  himself  after  an  abrupt  transition  to  a  strange  and 
mysterious  realm. 

For  a  similar  reason  the  basic 'idea  of  the  DilTerential  Calculus 
is  presented  in  a  study  of  the  derivative,  and  application  is  made 
to  some  of  the  simple  standard  functions.  Maclaurin's  formula  is 
also  obtained,  and  used  to  derive  several  standard  expansions, 
among  them  the  binomial  theorem  for  any  exponent. 

A  considerable  emphasis  has  been  placed  on  numerical  compu- 
tation, that  the  student  may  have  some  training  in  ready  calcula- 
tions. This  can  be  largely  supplemented  by  requiring  students 
to  work  out  mentally  in  class  many  of  the  numerical  exercises. 

It  has  been  thought  advisable  to  include  some  matter  which 
may  be  omitted  if  only  one  semester  is  to  be  given  to  this  course. 
Just  what  is  to  be  omitted  must  of  course  be  left  to  the  judgment 
of  the  instructor. 

W.  C.  B. 

Lincoln,  March,  1910. 


ADVANCED   ALGEBRA 

AND 

TRIGONOMETRY 


ADVANCED 
ALGEBRA  AND  TEIGONOMETRY 


CHAPTER  I 

The  Operations  of  Algebra 

1.  Letters  as  Symbols  of  Quantity.  —  In  algebra,  the  letters  of 
the  alphabet  are  used  to  designate  quantity  or  magnitude.  Thus 
we  speak  of  a  line  whose  length  is  I  feet,  of  a  weight  of  w  pounds, 
or  of  a  velocity  of  v  feet  per  second.  Here  the  letter  used,  I,  w,  v, 
is  suggested  by  the  quantity  considered,  length,  weight,  velocity. 
When  a  number  of  different  lines  are  considered,  say  n  lines,  their 
several  lengths  may  be  indicated  by  h,  h,  h,  •  .  -  ,  In,  or  by  l^^\ 
l^-\  l'^^\  '  •  '  ,  l'^^^-  Three  or  four  different  lengths  may  be  indi- 
cated by  accents  (called  "  primes  "),  as  V,  I",  V",  .... 

Fixed  or  known  quantities  are  usually  designated  by  the  first 
letters  of  the  alphabet,  as  by  a,  6,  c,  .  .  .  ;  unknown  quantities 
which  are  to  be  determined  from  given  data  are  represented  by 
the  last  letters  of  the  alphabet,  as  hy  x,  y,  z,  .  .  .  .  \i  x  denote 
a  quantity  of  a  certain  kind,  other  quantities  of  the  same  kind 
are  indicated  by  Xy,  X2,  X3,  .  .  .  (read,  "x  sub-one,  x  sub-two, 
X  sub-three,  etc."),  or  by  x^^\  x''-\  x^^\  .  .  .  (read  "x  super- 
script one,  X  superscript  two,  x  superscript  three,  etc."),  or  by 
x',  x'\  x'" ,  .  .  .  (read  "  x  prime,  x  second,  x  third,  etc."). 

2.  Signs  of  Relation.  —  These  are 

=  ,  read  "  equals,"  "  is  equal  to,"  etc.; 
5^,  read  "  is  not  equal  to  "; 


<f-read  ' 

'  is  less  than  "; 

> ,  read  ' 

'  is  greater  than  "; 

<,  read  ' 

'  is  not  less  than  "; 

>,  read  ' 

'  is  not  greater  than 

=  ,  read  ' 

'  is  identical  with  "; 

=  ,  read  "  approaches." 

V 

4  FUNDAMENTAL  OPERATIONS  [3 

Signs  of  Aggregation.  —  When  several  quantities  are  to  be 
treated  as  a  single  one,  they  are  enclosed  by  parentheses  (  ), 
brackets  [  ],  or  braces  \  j,  or  a  line  is  drawn  over  them,  called  a 
vinculum,  . 

Signs  of  Quality.  —  These  are 

+  ,  positive;  — ,  negative;  |  |,  absolute  value. 
The  first  two  simply  indicate  opposite  qualities;  thus,  if  -{-v,  or 
simply  V,  denote  a  velocity  in  one  direction,  then  —v  denotes  an 
equal  velocity  in  the  opposite  direction;  if  -\-t  denote  a  tempera- 
ture above  zero,  —t  denotes  an  equal  temperature  below  zero. 
The  third  symbol  is  used  to  indicate  that  we  are  dealing  simply 
with  the  numerical  (absolute)  magnitude  of  a  quantity,  without 
regard  to  its  sign. 

3.  The  Four  Fundamental  Operations.  —  These  are,  addition, 
subtraction,  multiplication  and  division,  indicated  by  the  symbols 
+>  ~>  X,  -r-,  respectively.  It  will  suffice  to  recall  the  rules  or 
laws  in  accordance  with  which  these  operations  are  to  be  performed. 
They  are  here  given  in  the  form  of  equations,  and  the  student  is 
asked  to  state  each  in  words. 

Laws  of  Addition. 

1.  If  a  =  6  and  c  =  d,  then  a  +  c  =  h  -\-  d. 

2.  If  a  =  6  and  c  7^  d,  then  a  +  c  7^  b  -\-  d. 

3.  a  -\-  b  =  b  -{-  a.     (Commutative  law.) 

4.  (a  -\-  b)  +  c  =  a  -{-  (b  -^  c).     (Associative  law.) 

Laws  of  Subtraction.  —  (Subtraction  defined  by  (a— 6)4-6=  a.) 


1. 

If  a  =  6  and  c 

=  d,  then  a  - 

-c  =  b-d. 

2. 

(a-c)-]-b  = 

(a  +  b)  -  c. 

3. 

a  +  (6  -  c)  = 

(a  +  6)  -  c. 

4. 

(a-hc)  -b  = 

(a-b)+  c. 

5. 

(a  -  c)  -b  = 

(a-b)-  c. 

6. 

a  -  (6  +  c)  = 

(a-b)-  c. 

7. 

a-(b-c)  = 

(a  -  6)  +  c. 

Laws  of  Multiplication. 

1. 

If  a  =  6  and  c 

=  d,  then  ac 

=  bd. 

2. 

If  ti  =  6  and  c 

7^  d,  then  ac 

7^  bd. 

3. 

aXb  =  b  X  a 

..     (Commutative  law.) 

4. 

aX  (b  X  c)  = 

(aXb)  X  c. 

(Associative  law.) 

5. 

(a-\-b-c)Xd  =  aXd  +  bXd-cXd. 

(Distributive  law.) 

4,5]  RATIONALITY.      ZERO  5 

Laws  of  Division.  —  (Division  defined  bj'  (a  -^  6)  X  6  =  a.) 

1.  U  a  =  b  and  c  =  d,  then  a  -^  c  =  b  -^  d,  provided  c,  d  j^  0. 

2.  {a  ^  b)  X  c  =  {aX  c)  ^  b,  provided      b  9^  0. 

3.  a  X  {b  ^  c)  =  (a  X  b)  -^  c,  provided      c  7^  0. 

4.  (a  ^  6)  -f-  c  =  (a  H-  0)  -^  6,  provided  6,  c  7^  0. 

5.  a  -^  (6  -^  c)  =  (a  -^  6)  X  c,  provided  b,  c  9^  0. 

Some  Working  Rules.  —  Tiie  sign  before  a  parenthesis  may  be 
changed  if  the  sign  of  each  of  the  terms  enclosed  is  changed  also. 

When  several  quantities  are  to  be  subtracted,  change  their  signs 
and  add  them. 

Division  may  be  expressed  as  a  multiplication  of  dividend  by 
reciprocal  of  divisor. 

The  sign  of  a  product  will  be  +  or  — ,  according  as  there  are 
an  even  or  an  odd  number  of  negative  factors. 

4.  Rational  Numbers.  —  All  positive  integers  can  be  formed  by 
adding  +1  to  itself  a  sufficient  number  of  times.  Through  the 
operation  of  subtraction,  negative  integers  are  introduced.  By 
performing  the  operations  of  addition,  subtraction  and  multipli- 
cation on  the  system  of  positive  and  negative  integers,  no  new 
numbers  are  formed.  Division,  however,  does  introduce  a  new 
class  of  numbers,  namely  fractions,  positive  or  negative,  formed 
of  the  quotient  of  two  integers. 

All  numbers,  positive  or  negative,  which  are  formed  of  the 
quotient  of  two  integers,  are  called  rational  numbers.  They  can  be 
obtained  from  + 1  by  means  of  the  four  fundamental  operations. 

Rational  Expressions.  —  Let  there  be  given  certain  quantities, 
a,  b,  .  .  .  X,  y,  .  .  .  .  Any  expression  which  can  be  built  up 
from  these  quantities  by  means  of  the  four  fundamental  operations 
is  called  a  rational  expression  (or  function)  in  terms  of  the  quan- 
tities involved. 

5.  Zero.  —  Zero  is  defined  as  that  number  ichich  may  be  added 
to  any  quantity  without  changing  the  value  of  the  quantity.  As  an 
equation,  the  definition  is 

a  +  0  =  a. 

Since  (a  -  0)  +  0  =  a, 

it  also  follows  that 

a  —  0  =  a. 


6  ZERO.     INFINITY  [6,7 

6.  The  operation  of  division  by  zero  is  excluded,  because,  what- 
ever be  the  number  a,  there  is  no  number  which  represents  a  -r-  0. 
The  reason  for  this  we  proceed  to  consider.  In  the  first  place, 
0  must  be  less  in  absolute  value  than  any  assignable  number, 
however  small.     For  if  this  were  not  the  case,  we  would  have 

a  +  0  5^  a.     Now  consider  the  quotient  r ,  and  suppose  a  to  be 

fixed,  and  6  to  be  taken  smaller  and  smaller.     As  b  tends  toward 

zero,  the  quotient  r  increases  without  limit  and   becomes  larger 

a 
than  any  assignable  number.     But  as  b  approaches  zero,  t  takes 

the  form  -  and  at  the  same  time  increases  without  limit  so  that 
no  value  can  be  assigned  to  this  form. 

Example.     Let  z  =  1. 

Then  x  =  a? 

(1  +  X)  (1  -  X). 


and 

1-X  =  1-X2=( 

Dividing  by  1 

—  X,  we  have 

1  =  1  +  X. 

Therefore 

1=2,    since  x 

We  are  led  to  this  fallacy  by  dividing  by  zero  in  the  form  of 
1  —  X.  Since  we  assumed  a;  =  1,  therefore  1  —  rr  =  0,  and  hence 
division  by  1  —  a:  must  be  excluded  in  this  problem. 

In  any  expression  involving  fractions,  those  cases  in  which  the 
denominator  of  any  fraction  vanishes  must  be  treated  as  exceptional 
and  especially  considered. 

If,  in  a  product,  a  factor  approaches  zero,  while  the  other  factors 
have  any  assigned  values,  then  the  product  approaches  zero. 
This  is  expressed  by  the  equation 

a  X  0  =  0. 

7.  Infinity.  —  A  quantity  which  increases  without  limit  is  said 
to  become  infinite.     When  b  =  0  ("b  approaches  zero  "),  if  a  is 

any  fixed  number,  r  increases  without  limit.     Such  quantities, 

which  are  larger  than  any  assignable  number,  are  all  indicated  by 
the  same  symbol,  oo  (read  "infinity").  As  an  example,  consider 
the  law  of  gases,  pressure  times  volume  is  constant,  or 

c 
pv  =  c,    or    p  =  -• 


8,9]  POWERS.     IMPORTANT  RELATIONS  7 

When  V  is  very  small  (relative  to  the  constant  c),  p  will  be  very- 
large,  and  as  v  becomes  still  smaller,  p  must  increase.  We  can 
choose  V  so  small  that  p  will  exceed  any  assignable  quantity,  or  p 
becomes  oo  when  v  =  0.     This  is  often  indicated  by  lim  i>  =  « 

r  =  0 

(read  "  the  limit  of  p  is  infinity,  when  v  approaches  zero  ")• 
We  are  thus  led  to  write  the  equation, 

—  =  QC,   when  «  9^  0. 

This  is  not  a  proper  equation,  but  simply  an  abbreviation  for  the 

statement,  "  A  fraction  whose  numerator  is  not  zero,  and  whose 

denominator  approaches  zero,  becomes  larger  than  any  assignable 

quantity." 

■    Since  a  quantity  which  increases  without  limit  can  be  made  as 

large  as  we  please  after  being  increased  or  diminished,  multiplied 

or  divided  by  any  number,"  we  have 

oo-|-a=co,  CO— a=co;  ooXa=oo,  co-^a  =  oo. 

'  8.  Powers.  —  For  brevity  we  put  a  X  a  =  a~,  a  X  a  X  a  =  a^, 
and  a  X  aX  a  ...  to  n  factors  =  a"".  The  quantity  a"  is 
called  the  nth  power  of  a.  The  number  n  is  called  the  exponent 
and  a  the  base  of  the  quantity  a". 

9.  Some  Important  Relations.  —  The  following  equations  and 
statements  should  be  verified  carefully  and  committed  to  memory: 

1.  (a  +6)2=  a2  +  2a6  +  62. 

2.  (a  -  6)2=  a?  -2  ah  +  62. 

3.  a--  62  =  (a  +  6)  (a  -  6). 

4.  a3+  6-^  =  (a  +  6)  {a^  -  ah  +  62). 

5.  a3-63  =  (a  -  6)  (a2  +  a6  +  62). 

6.  (a  +  6  +  c)2  =  a2  +  ^2  _^  c2  +  2  {ah  -^  ac  +  he). 

7.  The  square  of  any  polynomial  equals  the  sum  of  the  squares 
of  the  separate  terms  plus  twice  the  product  of  each  term  by  each 
following  term. 

8.  a"  —  6"  is  divisible  by  (a  +  6)  and  (a  —  6)  when  n  is  even. 

9.  a"  — 6"  is  divisible  by  (a  —  6),  not  by  (a  +  6),  when  n  is  odd. 

10.  a"  +  6"  is  divisible  by  (a  +  6),  not  by  (a  —  6),  when  n  is 
odd. 

11.  a''  +  6"  is  not  divisible  by  (a  +  6)  or  (a  —  6),  when  n  is 
even. 


8  .    EXERCISES  [10 

10.  Exercises.  —  Simplify,  by  removal  of  parentheses  and  col- 
lection of  like  terms: 

1.  ila-h)  +  a-ia). 

2.  il  a%  -  I  ab^)  +  (I  a%  +  |  ab^). 

3.  (0.8  a2  -  3.47  ab  -  17.25  ac)  -  (f  a^  -  0.47  ab  -  12t  ac). 

4.  (I  x2  +  3ax  -  I  a2)  -(2a? -ax-  la^). 

5.  Mx+  {[482/  -  (6z  +Sy-  7x)  +4z]-  [48y-8x+2z  -  (4x  +  y)]}. 

6.  6a  -  {4a -[86 -2a +  6] +  (36 -4a)}. 

Perform  the  operations  indicated  in  the  following  exercises  and  simplify  the 
results  when  possible: 

7.  -I  a^c  (I  62  -  4  c^  +  I  a(P  -  3). 

8.  3  xy"^  (x^  -3a?y  +  3  xy^  -  2). 

9.  0.6 ac^d'^  (2 arb  -  3cd^  +  hac^  -  5). 

10.  3i  a26c  (6  a^  -  4  62  +  2  a63  -  3  c2). 

11.  (x^  -2x  +  l){x^  -Sx+2). 

12.  (3  a^b  -  2  a%~  +  06^)  ( 2  a^  -  a6  -  5  62). 

13.  {x^  -lxY  +  ^  xy^  -  y^)  {x^  -2xif  +  y% 

14.  (a  +  bf  +  (a  -  bf. 

15.  {\a-^lf-{ha-l)\ 

16.  (x2  +  1  -  2/2  +  22/)  (^2  +  1  +  2/2  -  22/). 

17.  [ar  +  (a  +  6)  a;  +  a6]  [x-  -  (a  +  6)  .r  +  a6]. 

18.  [{x  +  o)2  -  a.r]  [{x  -  af  +  ax]. 

19.  a  (a  +  1)  (a  +  2)  -  (a  -  1)  (a  -  2)  (a  -  3) 

20.  [x{y-\)-y{x-  1)]  [(x  +  yf  -  (x  -  yf]. 

21.  31-^  ?/i^np5  _=.  _  io|  mhip^. 

22.  a26c7  ^  A  a462c8. 

23.  ifxV-^  -fia;V- 

24.  3  a2  (6  +  x)3  ^  6  a^  (6  +  x)^. 

25.  1.75  x^  (a;2  -  1)"  4-  25  x^  (1  -  a;2)2. 

26.  (8  a^6  -  24  0*6^  +  16  a768)  --  -  8  a%. 

27.  (8  xhj  -  ^  xy'  -  i  2/'  +  2  2/')  -  -  t  :^y^^ 

28.  x^  {or  +  62)  -  2  x*  (a?  +  b^f  ^  x^  (a2  +  62). 

29.  (6  ah  -  17  a2x2  +  14  ax^  -  3  x^)  -  (2  a  -  3  x). 

30.  (4  2/" -18  2/^ +  22  2/2 -7  2/ +  5)  -^  {2y-5). 

31.  [2  x»  +'7  xhj  -  9  2/-  (x  +  2/)]  ^  (2  X  -  3  2/). 

32.  (-Jd^  +  id*-nrf''+^)  -(-f^+2d). 

33.  (r\  a'-ia'b  +  U  ci%~  +  i  06^)  ^  (|  a  +  -J  6). 

34.  (-/j  m^  +  2^  to2^  _  2  5  ^^2  ^  1^4  „3)   ^  (1  ,„   _  7  ,j) 

35.  (x'^  -  §3  X*  +  U  x^  -  I  x2  -  \V  X  +  t)  H-  (x2  -  i  X  +  5). 

36.  (2«3 -.i6a  +  6)  ^  (a  +  3). 

37.  (4  x"  -  x22/2  +  6  X2/^  -  9  2/")  ^  (2  .r  -  X2/  +  3  2/2). 

38.  (x"  +  4  x22/2  -  32  y')  ^  (x  -  2  2/). 

39.  («^  -  5  «'''6'-  -  5  0=6^  +  6->)  -r  {a~  -  3  a6  +  62). 

40.  (x='-8  2/^)  -~{x-2y). 

41.  (A^'-O//)  -^(i.r+3  2/). 


11]  FACTORING  9 

42.  (27  a^b^  +  64  xV)  ^  (3  ab  +  ixy). 

43.  (a^b^  +  c»)  H-  (aV  _  abc  +^). 

44.  (m=^  -32y^)  ^  (ii-2v).  ^ 

45.  (a  -  6  +  c  -  d)-.  \    v> 

46.  (x  -  i  ?/  -  2  it  +  'f')-. 

11.  Factoring.  —  To  factor  an  expression  is  to  find  two  or  more 
quantities  wliose  product  equals  the  given  expression.  When 
two  or  more  expressions  contain  the  same  factor,  it  is  called  their 
common  factor. 

We  shall  illustrate  the  methods  commonly  used  in  factoring 
given  expressions  by  means  of  some  typical  examples. 

(a)  Expressions,  each  of  ivhose  terms  contains  a  common  factor. 
Example.     I  xhfz'  +  \  ^'iz  -  A  xVz'  =  i  x^-z  (h  xz^  +  J  -  }  x^)- 
(6)  Expressions  whose  terms  can  be  grouped,  so  that  each  group 
contains  the  same  factor. 

Example,     x'  -  7 x^y  +  Uxif  -Sy^  =  (x^  -Sy^)  -  (7 x-y  -  14 xif) 
=  {x-2ij){jr  +  2xy  +  4i-)  -7xy{x-2y) 
=.(x-2  2/)(x2-oxy/  +  4  2/2) 
=  {x-2y){x-y){x-4y). 

(c)  Trinomials  of  the  form  ax^  -{-bac  -^  c. 

Let  h,khe  a  pair  of  factors  whose  product  is  a,  and  m,  n  a, 
pair  whose  product  is  c.  Arrange  these  four  factors  as  in  the 
adjacent  schemes  ^X^^  ^X^  and  form  the  cross-products  as  indi- 
cated. The  sum  of  the  cross-products  must  equal  h.  If  this 
is  true  in  the  first  scheme,  the  factors  are  {hx  -\-  n)  {kx  -\-  m) ; 
in  the  second,  the  factors  are  (Jix  +  m)  {kx  +  n). 

Example.     12  x'  —  7  x  —  10. 

Here  h,  k  may  be  one  of  the  pcairs  of  numbers  1,  12,  or  2,  6,  or  3,  4,  both  num- 
bers to  be  taken  with  the  same  sign.  The  numbers  m,  n  may  be  —1,  10,  or 
+  1,  -10,  or  -2,  5,  or  +2,  -5.  By  trial  we  find  that  h,  k  must  be  3,  4, 
and  ni,  n  must  be  2,  —5.     The  factors  are  therefore  (3  x  +  2)  (4  x  —  5). 

To  find  the  factors  of  12  x^—  7  xij  —  10  y-,  we  would  proceed 
as  above  and  obtain  {3  x  -\-  2  y)  {4  x  —  5  ij). 

(d)  Expressions  ivhich  can  he  written  as  the  differetice  of  the 
squares  of  two  quantities. 

The  factors  are  the  sum  and  the  difference  of  the  two  quantities 
respectively. 

Example.         a*  +  a%-  -\-b^  =  a^  +  2  <rb"  +  /''  -  (^'b" 
=  («-  +  V'f  -  {abf 
=  (a^  +ab  +  b-)  (a^  -  ab  +  b"^). 


10  FACTORING  [12 

(e)  Expressions  of  the  form  P^ -\- 2  PQ  -\- Q- ,  where  P  and  Q 
are  monomials  or  polynomials. 

The  expression  is  then  the  product  of  two  factors  each  equal  to 
(P  +  Q),  and  is  therefore  (P  +  Q)-. 

Exam-pie.     x~  +  y-  -  2xy  -  4:ax  -\- 4:  ay  +4:  or 

=  (x  -y)-  -4a(x  -  y)  +4a2 
=  (x-2/-2a)2. 

(/)  Factor  Theorem.  —  If  a  polynomial  in  x  reduces  to  zero 
when  X  is  replaced  by  h,  the  polynomial  contains  the  factor 
(x  -  h). 

Proof:   Let  the  polynomial  be 

P  =  aox""  +  aix''-'^  +  aox""--  +  •  •  •  +  an-iX  +  an. 
Putting  h  for  x,  we  have  by  hypothesis 

ao/i''+ ai/i"-^ +  a2/i"-2  +  •  •  •  +  On-ih -\- a  n=  0. 

Therefore  by  subtraction, 

P  =  ao(x--  h-)+  adx^-'  -  h-^)+  aoix-^  -  h-^)+  •  •  • 
-{-an-\{x  -  h). 

But  each  term  of  the  right  member  of  the  last  equation  contains 
the  factor  x  -  h.  (See  8  and  9  of  9.)  Hence  P  is  divisible  by 
(X  -  h). 

Example.     Factor  x''  +  3  x^  -  4  x  -  12. 

If  this  is  the  product  of  three  factors  (x  -  h)  {x  -  k)  {x  -  I),  then  evi- 
dently hkl  =  12.  Hence  we  substitute  in  the  given  polynomial  the  factors 
of  12,  and  find  that  it  vanishes  when  x  =  2,  x  =  -  2,  and  x  =  -  3.  Hence 
the  factors  are  (x  -  2)  (x  +  2)  (x  +  3). 


12.   Exercises.  —  Factor 
1. 


'       ^L  -^-.  11.  :,^  _  X  -  110. 


2«      8«'       4a^  j2.  7  +  10x  +  3.c2. 

2.   x2-2ax  +  o2-T/2.  j3^  x'-lOa^x  +  Oa^ 

4.  15x2 -7x- 2.  ^^  xV-3x?/z-10A 

5.  6x''  +  lQxy-7y\  ^g.  2  +  7x-15x2. 

6.  x2-2x-24.  ^^  x3-G4x-x2  +  64. 


7.   8x''-27xz; 


18.   oS  +  1. 


8.   27x*+Sxy^.  ^q    ^6  _  j 

'       13/ +  36.  20:    (a+bf  +  l. 


9.    X 

10.    4  a"  -  5  a2  +  1 


13,14]  H.C.F.  AND  L.  C.  M.  11 

21.  (x^  +  v")  -  (X  +  yf. 

22.  «■»  +  b''  -  c*  -  d^  +  2  a'^lf-  -  2  c^^. 

23.  d'c-  +  acd  +  afec  +  bd. 

24.  1  -  aV  _  62^2  ^  2  afexy. 

25.  xV  -  x^y^  -  x^if  +  xy*. 

26.  a8-82a'*  +  81. 

27.  xV-17x2y-110. 

28.  (a2+3)2-36a2. 

29.  x'''+9x2  +  16x  + 

30.  4x'  -Sx'''  -X-  +  ^ 

13.  Highest  Common  Factor.  —  The  highest  common  factor 
(H.  C.  F.)  of  two  or  more  polynomials  is  the  polynomial  of  highest 
degree  that  will  divide  them  all  without  a  remainder. 

When  each  of  the  given  polynomials  can  be  factored  by  inspec- 
tion, the  H.C.F.  is  easily  determined  from  their  common  factors. 
Example.     The  H.  C.  F.   of  32  (x  -  l)2(x  +  l)3(x2  +  1)   and   24  (x  -  1)3 

(X  +  I)2(x2  +  1)2  is  8  (X  -  1)2(X  +  l)2(x2  +  1). 

When  the  given  polynomials  cannot  be  readily  factored,  we  use 
a  method  like  that  of  arithmetic. 

Let  the  given  polynomials  be  Pi  and  Po  and  let  Q  be  the  quo- 
tient and  R  the  remainder  when  Pi  is  divided  by  P^.     Then 

Pi  =  PiQ  +  R. 
Hence  any  factor  common  to  Pi  and  P2  is  also  a  factor  of  R. 
Hence  it  is  a  common  factor  of  P2  and  R.     Divide  P2  by  R, 
obtaining 

P2  =  RQi  +  Rx. 
Hence  a  common  factor  of  P2  and  R  is  also  a  factor  of  Pi.     Divid- 
ing R  by  Pi,  we  obtain 

R  =  R\Q2  +  P2, 
and  the  common  factor  must  be  present  in  P2,  and  so  on. 

Ride.  —  If  at  any  step  there  is  no  remainder,  the  last  divisor 
is  the  required  H.  C.  F. 

14.  Least  Common  Multiple.  —  The  least  common  multiple 
(L.  C.  M .)  of  two  or  more  polynomials  is  the  polynomial  of  lowest 
degree  that  is  exactly  divisible  by  each  of  them. 

When  the  given  polynomials  can  be  easily  factored  by  inspec- 
tion, form  the  product  of  all  the  types  of  factors  present  in  any 
of  them,  taking  each  factor  the  greatest  number  of  times  that  it 
occurs  in  any  of  the  given  expressions;  this  product  is  their  L.  C.  M. 


12  H.C.F.  AND  L.C.M.  [15 

When  the  given  polynomials  cannot  readily  be  factored,  their 
L.C.M.  is  obtained  by  use  of  the  following  theorem: 

The  product  of  the  H.  C.  F.  and  L.  C.  M.  of  two  polynomials 
equals  the  product  of  the  polynomials. 

Proof:  Let  F  be  the  H.  C.  F.,  and  M  the  L.  C.  M.  of  the  two 
polynomials  Pi  and  P2.     Also  let 

^  =  Qi  and   ^  =  Q2; 

then  Pi  =  FQi    and   Po  =  FQ2. 

Hence  P1P2  =  FX  PQ1Q2. 

Since  F  contains  all  factors  common  to  Pi  and  P2,  Qi  and  Q2 
have  no  common  factor,  and  the  product  FQ1Q2  contains  all  the 
factors  of  the  types  present  in  both  Pi  and  P2. 

.-.    M  =  FQ1Q2  =  ^^;     or,     MF  =  P.P.. 

Rule.  —  To  find  the  L.  C.  M.  of  two  polynomials,  divide  their 
product  by  their  H.  C.  F. 

To  find  the  L.  C.  M.  of  more  thai>  two  polynomials,  find  the 
L.  C.  M.  of  two  of  them,  then  the  L.  C.  M.  of  this  and  a  third  one 
of  the  polynomials,  and  so  on. 

15.  Exercises.  —  Find  the  H.  C.  F.  of 

1.  6  (x  +  If  and  9  (x^  -  1). 

2.  a"  -  6"  and  a''  -  6". 

3.  12  (x2  +  2/2)2  and  8  (.r"  -  1/). 

4.  u^  —  v^  and  vr  —  v". 

5.  {c?x  —  ax"f  and  ax  {a?  —  x^f. 

6.  27  (a"  -  h^)  and  18  (a  +  hf. 

7.  (24  a^  _i-  36  ab  -  48  ac)  and  (30  a?  +  45  a%  -  60  a\). 

8.  125  x^  -  1  and  35  x^  -  7  a:  +  5  ax  -  tt. 

9.  4x2- 12x1/ +9?/2  and  4x2 -9?/2. 

10.  x2  +  2x  -  120  and  x2  -  2x  -  80. 

11.  12x2 -17ax  +  6a2  and  9x2  +  6ax -8a2. 

12.  x3  +  4  x2  -  5  X  and  x3  -  6  X  +  5. 

13.  x^  +  3  x2  +  7  X  +  21  and  2  x"  +  19  .r  +  35. 

14.  0^  +  703+702  -  15o  and  n^  -2a^  -  13a  +  110. 

15.  20x''  +  x2  -  1  and  75  x"  +  15  x^  -  3  x  -  3. 

16.  x"  -  ox'  -  a2x2  -  o^x  -  2  o"  and  3  x^  -  7  ax-  +  3  a'x  -  7  a^  . 

17.  x^  -  y\  x^  +  ?/,  and  x^  +  y^. 

18.  x2  -  2  a2  -  ax,  x2  -  6  a2  +  ax,  and  x2  -  8  a2  +  2  ax. 

19.  o^  +  02^2  +  b\  a*  +  ob^,  and  a%  +  ?>^. 

20.  3x3  -  7x2^/  +  5x1/2  -  y3^  ^2y  _|_  3x2,2  _  3x3  -  yS^  and  3x3  +  5  x22/  + 
X2/2  -  2/3. 


16,17]  FRACTIONS  13 

FindthcL.C.  M.  of:      ■ 

21.  S  a-. r-y^  and  12  ahx'y-. 

22.  4ax^y',  Gn^xy^,  and  ISa^x-y. 

23.  a^  -  b^  and  (a  -  b)~. 

24.  cv^bx  —  abhj  and  abx  +  Iry. 

25.  .r-  -  3  X  -  4  and  x'  -  x  -  12. 

26.  x2  -  1  and  x2  +  4x+3. 

27.  6x2  +  5 X  -  6  and  6 x^  -  13 x  +  6. 

28.  12x2  +  5x  -  3  and  6x='  +  x^  -  x. 

29.  12  x2  -  17  ax  +  6  a^  and  9  x^  +  6  ax  -  8  a^. 

30.  a3-9a2+23a-15  and  a2_8a  +  7. 

31.  ??i^  +  2  m^n  —  y^m^  —  2  n^  and  ?«^  —  2  ??r-/t  —  ?/;/r  +  2  n'. 

32.  x^  —  ir,  (x  —  ?/)2,  and  x  -\-y. 

33.  x^  +  3  X  +  2,  x^  +  4  X  +  3,  and  x"  +  5  x  +  6. 

34.  X"  +  5x  +  10,  x^  -  19x  -  30,  and  x^  -  15  x  -  50. 

35.  x^  +  2  X  -  3,  x^  +  3  x2  -  X  -  3,  and  x^  +  4  x-  +  x  -  6. 

36.  6x-  -  13x  +  6,  6x2  +  5.^  _  6,  and  9x2-4. 

37.  x2  -  1,  x2  +  1,    and  .t3  +  1. 

38.  x2  +  1,  x""  -  1,  and  x"  -  1. 

39.  a^  -  6^  a9  -  b^,  and  a^  -  b^. 

40.  x2  -  i/2,  x^  +  y\  x3  -  y^,  and  x^  +  7/. 

16.  Fractions.  —  An  algebraic  fraction  is  the  indicated  quotient 

of  two  algebraic  expressions.     It  is  written  in  the  form   y,,  N 

being  called  the  numerator  and  D  the  denominator. 

When  A''  and  D  have  a  common  factor  F,  so  that  we  may  put 
A^  =  NiF  and  D  =  D^F, 
then  the  fraction  may  be  simplified  as  follows:  ^ 

N  ^NiF  ^N, 
D      DxF      Di' 
When  all  factors  common  to  A^  and  D  have  been  removed  in 
this  way,  the  fraction  is  said  to  be  reduced  to  its  lowest  terms. 

When  the  common  factors  of  N  and  D  are  not  obvious  on 
inspection,  find  the  H.C.F .  oi  N  and  D,  and  remove  it  as  above. 

17.  Sign  of  a  Fraction.  —  By  the  rules  for  division  we  have, 

N  ^_  ^^  ^_  Ji     ^  ZL^ 

D~      D  -D-d' 

Hence  the  rules:  Changing  the  sign  of  either  numerator  or  denomi- 
nator changes  the  sign  of  the  fraction. 

Changing  the  signs  of  both  numerator  and  denominator  does  not 
affect  the  sign  of  the  fraction. 


14  FRACTIONS  [18 

The  sign  of  a  fraction  may  be  changed  either  by  changing  the 
sign  standing  before  the  fraction,  or  by  changing  the  sign  of  the 
numerator  or  of  the  denominator. 

18.  An  integral  expression  is  one  whose  literal  parts  are  free 
from  fractions. 

A  mixed  expression  is  one  formed  from  the  sum  of  an  integral 
part  and  one  or  more  fractions. 

A  complex  fraction  is  one  whose  numerator,  or  denominator, 
or  both  are  fractions  or  mixed  expressions. 

Every  mixed  expression  and  every  complex  fraction  can  he  re- 
duced to  a  simple  fraction  {or  to  an  integral  expression). 

For,  two  or  more  simple  fractions  can  be  reduced  to  a  common 
denominator  and  then  combined  into  a  single  fraction  by  writing 
the  sum  of  the  numerators  over  the  common  denominator.  For 
this  purpose  the  simplest  common  denominator  is  the  L.  C.  M.  of 
the  separate  denominators.  This  is  called  the  least  common  de- 
nominator of  the  fractions  considered.     In  this  manner  we  reduce 

D[^D,^  ^"^  D 

A  mixed  expression  is  reduced  by  the  formula 
N      PD  +  N 
^'^  D~        D 
Finally,  a  complex  fraction  is  reduced  by  first  reducing  its 
numerator  and  denominator  separately  to  simple  fractions.     The 
reduction  is  then  completed  by  the  formula, 
N 


D       N      D'      ND' 

W~  D^N'      N'D ' 

D' 

Examples. 
1.   Simplify 

X      y             X             y 

First  reduce  each  fraction  to  a  simple  fraction,  thus: 

2               2            2X2/ 

1      1       y  -X       y-x 

X      y         xy 

y              y             xy 
2/       x-y       x-y 

X                 X 

19]  EXERCISES  15 

__x ^  _      X        ^  _     xy    ^      xy 

1  _  5  ?/^_^  y  -  X     X  -  y 

y  y 

Reducing  to  the  common  denominator  x  —  y,  we  liave 

^^  +  _^  +  ^^  =  Ill^^Jl£^±f2/  =  0  (provided  X  ^  2/). 
y -x '    x -y      x -y  x-y 

o       ,  x2  ,  x2  ,  x2 

2.     X3 ; =  X3 


1   -  X4  ,1  -  X4  X  (1  -  X4) 

X  X 

a:2  ,         x2  ,  ,   1       x4  + 1 

X3 r  =  X3  + 


X  —  X  (1  +  X2)  —  X3  X  X 

19.   Exercises.  —  Reduce   to  simple   fractions  or    to    integral 

expressions: 
•      W^±f_«^^\(a2_:,2)  n.   x^+y_2^3x2+3y2^ 

^-    U  -  ^      «  +  -^i  ■^^-    x2  -  2/2  •       x  +  y 


2    "'~^%"'~^'.                               1„    45  (x  -  2/)  .     27(x-y)2. 
a3  +  63         2  a6  12. -^ 

a4_-64      x2J^xy_+y_2 

x3  -  7/3  ^       a2  +  62      •  13. 

^-  (M)(^^)- 

/x5  _  7/\        /x  _  2/\ 
xi2  +  yi2    _   x4  +  2/4 


14. 


16. 


xi2  -  2/12       x^  —  2/8 


32(3+2/) 

•   1286(z+2/)2 

a2-4x2 
a2  +  4  ax 

rt2  _  2  ax 
■  ax  +  4  x2 

a2  +  a6  . 
a2  +  62    • 

ah  (a  +  6)2 
a*  -  64 

t|3  -  1,3 
Ulv2  -  ui 

m2  4-  liv  +  ti2 

p2  +  3  p  +  9  .  p3  _  27 

x  +  1   ,  y  +  1  ,,,     x6  -  2/^  ^  x2  +  x?/  +  2/2 

(X  -  2/)2  •  X 

X  -  2/      x3  -  2/3 


_1_  ^_JT-/  J7_ 

2/  '    (a;  -  2/)2  X  -  2/ 


11 

X     y 


-o    a:  +  2/      x3  +  2/^ 
9.    1+^-  '   x  +  2/      X2+2/2' 

x+-  X  -y      x^  -yi 

X 

-4-+-^-  19. — ^ — 

a: y_  1    I   1  +y 

x-  y      x  +  y  3-2/ 

1.2.1 
20 


21. 


X  (X  -  1)     '     1   -  X2    '    X  (X  +  1) 

1  2,1 


x2-5x  +  6      x2 -4x  +  3  '  x2-3x  +  2 


16  EXERCISES  [19 

22.       "       I       ^       I      ^^' 

'    U  +V       U  —  V       U'  +v^ 

-  1  2  (x  -  2)      ,         x-3 


23. 
24. 


a;2-5x+6      x2  -  4x  +  3  '  a;2  -  3x +2 
7+3x2         5-2x2         3  -2x+x2 
4-x2       4+4x+x2      4-4x+x2' 


l-2x  2x-3     ,  1 


3  (x2  -  X  +  1)      2  (x2  +  1)   '  6  (x  +  1) 
6c      ac      a6      a/  \         a  +  b  +  cj 


27.    /^!  +  «^-?-«  +  l 


28. 


\a      X      6      ?//  Va      S      ^      2// 

29    (l-  ^^"1    /7x        49x2        343x3  \ 

29-   1,^      ir^j    ini/  + 121 2/2  + 1331  W 

(ah       3bc\  /5ac      7abW3b  _  ah  \ 
V3c2      5a2J  \762       902^  Uo2      3c2^ 
M x3j/2  _  3x2j/3      2xy4  _  ?/\  /2x2y  _  3xy2  _  3y3\ 
^      i  5a3         2a26  +3a62      h^)\Za-i        5 ah       2h'^l' 


CHAPTER  II 
Involution.     Evolution.     Theory  of  Exponents.    Surds 

AND    ImAGINARIES 

20.  Involution  is  the  operation  of  raising  a  quantity  to  an 
indicated  power. 

The  symbol  a^  represents  a  X  a  X  a  ...  to  n  factors  (8), 
n  being  a  positive  integer.  Hence,  if  m  be  a  second  positive 
integer,  we  have  by  cancellation, 

(1)  —  =  a"-'"  when  n  >  m; 

(2)  —  =  ^—  when  n  <m. 

a'"      a'"-" 

Negative  Exponent.  —  We  now  defijie  the  symbol  a-"  to  be 


a^      a  X  a.  X  a  ...  to  n  factors 
Then  — ^  =  a-^'"-")  =  a"-"». 

We  may  now  write, 

(3)  ^  =  a"-'", 

a"* 

whether  n  is  greater  or  less  than  m.  Hence  by  the  introduction 
of  the  negative  exponent,  the  two  equations  (1),  (2),  may  be  written 
as  a  single  equation,  (3). 

We  now  easily  verify  the  following  rules  for  operating  with 
integral  exponents,  positive  or  negative. 

IV.   (a"')"=a""\ 

V.    (ab)"=  a"b'\ 

17 


I. 

1 

a~    =  - — 

11. 

rt"  X  a"'  =  ft' 

III. 

a"  -^  a'"  =  a' 

18  INVOLUTION.    EVOLUTION  [21,22 

21.  Exercises. 

1.  State  the  above  rules  in  words. 

2.  Verify  the  above  rules  by  means  of  the  definitions  for  a"  and  a  "". 

3.  Show  that  rule  II  contains  rule  IIL 

4.  Show  that  rule  V  contains  rule  VL 

Perform  the  operations  indicated  in  the  following  exercises,  and  express 
the  results  in  forms  free  from  fractions: 

6.    (^5)'.  8.     [(ax)3m  +  4.]5^-6n. 


ax^ 


(a263)3"  Xd^x^J     '  \  6c4 


)'■ 


Zero  E:q)onent.  —  If  in  rule  III  we  put  n  =  m,  we  get 


But  a"-^  a'^=  1.  Therefore  we  define  the  symbol  a^  by  the 
equation  a^  =  1.  Then  III  is  true  fpr  all  integral  values  of  n 
and  m,  equal  or  unequal.     Hence  we  add  to  the  above  rules: 

VII.   ao  =  1.  1 

22.  The  nth  root  of  a  quantity  a  (symbol  ^/a  or  a")  is  a  quan- 
tity whose  nth  power  is  equal  to  a. 

Evolution  is  the  operation  of  finding  the  indicated  root  of  a 
quantity. 

By  definition,  we  have 

y/aX\/aX^a   ...  to  n  factors  =  (7a)'*=  a, 

111  /    l\n 

or  a"  X  a"  X  a"  ...  to  n  factors  =  \  a"/    =  a. 

The  last  equation  will  be  covered  by  rule  IV  (20)  if  we  extend 
that  rule  to  the  case  where  m  is  the  reciprocal  of  a  positive  inte- 
ger. We  now  extend  rules  I- VI  and  asswme*  that  m  and  n  may 
be  not  only  positive  or  negative  integers  or  zero,  but  also  the 
reciprocals  of  positive  or  negative  integers. 

If  we  let  n  =  -  and  m  =  - ,  r  and  s  being  integers,  we  have 


23, 24  ] 


INVOLUTION. 

EVOLUTION 

V 

V.a-l-l- 

/   i\i        1 

IV'.   Wy=a''. 

CL^ 

1         1   1 

1, 

1            1             1^1 

v.    (aby  =a'h\ 

\ 

II'.  or  Xa'  =  a'-    «. 

1        1 

1         1         11 

vi'-(;:y='^- 

Iir.  or  ^a'  =  a"-    ^ 

h^ 

These  equations  define  the  rules  governing  operations  involving 
roots. 

Exercise.  State  the  above  rules  in  words.  What  is  the  meaning  of  a 
negative  root? 

23.  Rational  Exponent.  —  By  the  preceding  laws  we  now  have 
a  meaning  assigned  to  the  symbol  a^  when  n  is  any  rational 
number  (4).     For,  if  n  =  p  -^  g,  p  and  q  being  integers,  we  have 

E    [  ly  I 

a^  =  a''=[a'il  =  (qp)"  ; 

E 
that  is,  a''  means  the  pth  power  of   the  gth  root  of  a,  or  the  5th 

root  of  the  pth  power.     In  a  fractional  exponent,  the  numerator  is 

the  index  of  the  power,  the  denominator  the  index  of  the  root. 

By  combining  rules  I-VI  and  I'-VF,  we  see  that  the  former 

set  of  rules  holds  when  m  and  n  are  any  rational  numbers.     Hence 

we  adopt  the  rules  of  (20)  as  the  rules  governing  quantities  affected 

with  rational  exponents. 

24.  Irrational  Numbers.  —  By  the  operation  of  evolution  we 
are  led  to  numbers  which  cannot  be  produced  from  integers  by 
means  of  the  four  fundamental  operations.  Thus  if  we  attempt 
to  calculate  \^2  we  are  led  to  a  non-terminating  decimal.  To 
four  decimals  Ave  have 

1.4142  <  \/2  <  1.4143, 

or  14142   .  w^o  ^  ^^^ 

10000  10006 

We  have  here  two  rational  numbers  between  which  V2  lies.  By 
going  out  to  a  sufficient  number  of  decimals,  we  can  obviously 
obtain  two  rational  numbers  containing  V2  between  them  and 
differing  from  it  by  as  little  as  we  please.  By  taking  successively 
4,  5,  6,   .    .    .   decimals,  proceeding  as  above  and  noting  each 


20  INVOLUTION.    EVOLUTION  [25 

time  the  smaller  of  the  two  rational  numbers,  we  obtain  a  series 
or  sequence  of  rational  numbers  which  increase  and  approach 
V2;  by  noting  each  time  the  larger  of  the  two  numbers,  we  obtain 
a  second  sequence  of  rational  numbers  which  decrease  and  also 
approach  V2. 

If  on  the  other  hand  we  consider  the  sequence  of  numbers 

13     133     133J5 

'  lO'    lOO'    lOOO' 

4 
these  evidently  approach  the  value  ^ '  which  is  a  rational  number. 

The  idea  here  indicated  is  used  to  define  irrational  numbers. 
Without  going  further  into  the  subject  here,  we  shall  say  that  an 
irrational  number  is  one  which  can  he  represented  to  any  degree  of 
approximation,  hut  not  exactly  as  the  quotient  of  two  integers. 
Such  numbers  may  be  produced  in  performing  the  operation  of 
evolution  on  rational  numbers. 

Real  Numbers.  —  The  rational  numbers,  including  all  integers 
and  quotients  of  integers,  and  the  irrational  numbers  together 
constitute  the  class  of  real  numbers. 

Irrational  Expressions.  —  We  now  extend  the  idea  of  irration- 
ality to  algebraic  quantities  in  general  by  the  following  definition: 

An  algebraic  expression  is  said  to  he  irrational  when  its  parts 
are  affected  by  other  than  the  four  fundamental  operations. 

Hence  any  expression  involving  indicated  roots  is  irrational.  As 
examples,  we  have 

^ /l  -}..  2 

VI  +  X-;  (x--  xy)  -'  +  (xy  -if);  y  j- 


1  +2a  +  a- 
The  last  expression  may  be  simplified.     Thus, 


\/ 


l+2a+a2      Vl  +  2a4-a2        1  +  a 


Vl-a  Vl 


A  surd  expression  is  one  involving  an  indicated  root  which  can- 
not be  exactly  found. 

A  surd  number  is  an  indicated  root  of  a  number  which  cannot 
be  exactly  found. 

25.  Irrational  Exponents.  —  What  meaning  shall  we  attach  to 
the  expression  2"^-?    Let  ai,  ao,  as,  ...  be  a  series  of  rational 


I. 

1 

II. 

(V^iO'  =  (v' 

[II. 

«*  -^  rt"  = 

26]  IMAGINARY   NUMBERS  21 

numbers  approaching  V'l  in  value.     Then  the  quantity  toward 
which  the  scries  of  numl^ers  2«',  2"2,  2'*%  .  .  .  approaches  is  2^-. 
Similarly  we  obtain  a  meaning  for  a^,  when  x  is  irrational. 
We  now  define  a^  as  a  symbol  subject  to  the  following  laws: 

rV.    («')"=  (V^»  {nota"^"); 
V.    {ahY=  a^'h''; 

VI.(^)'=$; 

provided  that  the  symbols  a,  b,  x,  y,  a'',  h'',  a^,  W  stand  for  real 
numbers. 

26.  Imaginary  Numbers.  —  When  x-  =  1 ,  we  have  obviously 
X  =  ±  I.  What  is  X  when  x^  =  —  1?  The  answer  cannot  be 
a  real  number,  since  the  square  of  every  such  number  is  posi- 
tive. To  obtain  an  answer  to  the  question,  we  introduce  a  new 
number  whose  symbol  is  V  —  1,  and  which  is  defined  as  the 
cjuantity  whose  square  is  —1. 

Since  V—  1  is  not  a  real  number,  it  is  often  called  imaginary 
and  denoted  by  i.  Hence  the  quantity  i  =  V  —  lis  defined  by  the 
equation  i"^  =  —  1.      

We  now  define  V  —  a  by  the  equation 

I.  V^^  =  i  Va. 

(This  is  in  accordance  with  our  rules  for  exponents,  since 

V^  =  VaX-l  =  Va  V  -  1  =  i  Va.) 
Then  the  product  V^^  xV-  bis  determined  by  the  equation, 
n.      \/^^  X  V^^  =  iVaXiVb  =  i^  Vab  =  -  Vab. 

The  results  of  the  operations  of  algebra,  applied  to  any  number, 
are  always  expressible  in  the  form  a  +  hi,  where  a  and  b  are  real. 
Such  a  result  may  be  considered  as  consisting  of  a  real  units  and 
b  imaginary  units,  «  X  1  +  &  X  *;  it  is  called  a  complex  number. 

Two  numbers  of  the  forms  a  +  bi  and  a  —  hi  are  called  con- 
jugate complex  numbers. 

When  a  =  0,  the  complex  number  a  +  6i  becomes  hi  called  a 
pure  imaginary. 


22  SURDS  [27,28 

The  rules  for  operating  with  complex  numbers,  aside  from  II 
above,  are  considered  in  chapter  17. 

Principal  Root.  —  There  are  in  general  n  distinct  quantities,  the 
nth  power  of  each  of  which  equals  a  given  number  a  (see  259). 
That  is,  a  given  number  has  in  general  n  distinct  nth  roots. 
Thus, 

the  square  of  +  2  or  —  2  is  4; 

the  cube  of  -  2,  (1+  i  VS),  or  (1  -  iVs)  is  -  8; 

the  fourth  power  of  -\-  2,  —  2,  -\-  2  i  or  —  2  i  is  16. 

The  principal  root  of  a  number  is  its  real  positive  root  when 
one  exists;  if  not,  its  real  negative  root;  when  all  roots  are  imagi- 
nary, any  one  of  them  may  be  chosen  as  the  principal  root. 

1 

In  this  text  the  symbol  for  a  root,  y/a  or  a'*,  will  mean  the 
principal  root  only. 

Thus:  V4  =  2,  not  ±  2;  if  we  wish  to  indicate  both  square  roots, 
we  always  write  ±  Va. 

27.  Reduction  of  Surds.  —  The  expression  'l/a  is  usually  called 
a  radical,  V  being  the  radical  sign,  n  the  index  of  the  radical  and 
a  the  radicand.  When  the  radicand  is  not  a  perfect  nth  power, 
the  expression  is  a  surd. 

A  surd  is  said  to  be  in  its  simplest  form  when  all  factors  of  the 
radicand  which  are  perfect  powers  of  the  same  index  as  that  of 
the  radical  have  been  taken  out  from  the  radical  sign.     Thus: 


s/^ 


Sa*b^      2ab  ,,-^ 
27?  =3^  -J^bK 


Two  surds  are  similar  when  they  can  be  expressed  with  the 
same  index  and  radicand.     Otherwise  they  are  dissimilar. 

A  quadratic  surd  is  one  whose  index  is  2. 

28.  The  sum,  difference,  product  and  quotient  of  two  dissimilar 
quadratic  surds  are  always  surds. 

Proof:  Let  the  surds  be  Va  and  Vb.  Since  they  are  dissimilar, 
neither  ah  nor  a  -j-  6  can  be  a  perfect  square.  Hence  the  product 
or  quotient  of  the  two  surds  is  a  surd. 

Further,  let  c  be  a  rational  number,  and  assume  that 

Va  ±  Vb  =  c. 


29-32] 

SURDS 

Squaring, 

a  ±2  Vab  +  b  =  c, 

or 

±  2  Vab  =  c  -  a 

23 


b. 

But  a  surd  cannot  equal  a  rational  expression  by  definition. 
Hence  the  assumption  is  false,  and  the  sum  or  difference  of 
two  surds  is  also  a  surd. 

29.  Given  a  relation  of  the  form  a  -]-  V'b  =  c  -{-  Vd;  then  a  =  c 
and  b  =  d. 

For,  on  transposing,  we  have  ^b  —  Vd  =  c  —  a;  hence  \i  b  9^  d, 
we  have  a  surd  equal  to  a  rational  number,  which  is  impossible. 
Therefore  b  =  d.     Hence  also  a  =  c. 

30.  To  rationalize  the  denominator  of  —7= -=  • 

Va  +  Vb 

Rule.  —  Multiply  both  sides  of  the  fraction  by  Va  —  Vb. 

p^^  A  {Va  +  Vb)  ^  A{Va+Vb)  ^ 

(Va  +  V6)  Wa  -  Vb)  a-b 

31.  To  obtain  the  square  root  of  a  -{-  Vb. 

Assume  that    V  a  +  Vb  =  Vx  +  Vy.     To  find  x  and  y. 
Squaring,  a  +  Vb  =  x  -\-  y  -\- 2  Vxy  =  x  -{-  y  -\-  V^  xy. 

Hence  a  =  x  +  y  and   b  =  4xy     (29). 

Then  a^  —  b  =  x~  —  2xy  -\-  y-  =  {x  —  y)'-, 

or  ±  Vd"  —  b  =  X  —  y. 

But  a  =  a:  +  2/. 

Therefore  x  =  \  (a  ±  Vd-  —  b)  and  y  =  ^  {a  T  Va^  —  b). 

32.  The  index  of  a  surd  may  be  multiplied  by  any  number  if  at 
the  same  time  the  radicand  be  raised  to  the  power  indicated  by  this 
number. 

_  1  m_  

For,  ^a  =a''  =  a'""  =  "'7«"'- 

In  combining  surds  by  multiplication  or  division  this  rule  is 
used  to  reduce  them  to  surds  with  a  common  index.  This  is 
accomplished  by  writing  all  the  surds  as  fractional  exponents  and 
then  reducing  the  exponents  to  a  common  denominator. 


24  EXERCISES  [33 

33.   Exercises. 

Write  the  following  with  positive  exponents  and  in  simplest  form : 
3a0  6-2c-4 


1. 


4. 


X 

x5yS 

■1 

/    a-? 

\- 

15 

L*2/-3 

] 

/      .i 

^J 

-12n_ 

la-i6- 

/r-?- 

V 

-iN-2 

Reduce  to  radicals  with  the  same  index  : 

9.   V3    and  -^l.  19.  Va,  'X/^.  and  -sjc. 

10.  -732  and  <J^.  20.  V-^^  and  %/x5. 

11.  -^2    and  ^/3.  21.  </^,  </a3,  and  -^S. 


12.  y/S    and  v25.         |  svj 

13.  V5,  ^/2,  and  -s/S.'  ■  ■■■^' 

14.  V3,  V8,  and  ^4.  23.   y/j,     y/^ ,  and  0^. 

15.  -^i  -s/f,  and  Vf  •  ,  , 

16.  -s/i,   VA,  and  7FiI-  2*'    V 

17.  \/l  Vf,  and  '-n/A- 

18.  ■\/(h3,  -s/l,  and  S/lT- 


1  /—  ,  and  i  / 

t /4;,  and  \/-- 
_       V  2/-  V_2 

25.    i'/i^,      V.r,  and  (/^ 


Combine  by  performing  the  indicated  additions  and  subtractions,  reduc- 
ing to  similar  surds  when  necessary : 

26.  2  V3  -  5  V3  +  9  V3.  32.  2  Vl75  -  3  V<i3  +  5  V28. 

27.  4  %/4  -  3  ^/4  +  2  ■^.  33.  3  V"  +b^a  -  V«^- 

28.  3  -V^  +  \/32.  34.  \/2Y7i  -  \/,S^  +  \/64c8. 

29.  \/2  +  3  V32  -  ^  Vl28.  35.  VfAc  -  V"^  +  Vo^- 

30.  5  \/4  +  2  -\/32  -  </T()8.  36.  ^o^  +  V^^^^^  -  "^  V^^- 

31.  ^  ^/5  +  2i  </5  +  i -V-iO. 

Reduce  to  the  form  \].x  +  V// : 


37. 

V4  +  2  V3. 

38. 

V3  +  V5- 

39. 

V2  +  V3. 

40. 

Vs  +  Vis. 

42. 

Vio-f 

-  2  V2T. 

43. 

V7  + 

2  Vio. 

44. 

Vt- 

4  V3. 

45. 

Vl3- 

2  V30. 

41.    Vs  -  V21.  46.    Vu  -  4  V7. 


33] 


EXERCISES 


25 


Perform  the  following  multiplications  uiul  divisions : 


47.  (3+2V2)  (3-2  V-^)- 

48.  (5+2V3)(3-5V3). 

49.  (2V6-3V5)(V;5  +  2Vli). 

50.  (V7-  V3HV2  +  Vo)- 

51.  (V9  -  2  V4)  (4  -^  72). 
62.  (  Va +6  +  V« )  (V" + 6  -  V'O  ■ 
53.   Vm  +  V"  ^'"  -  V"- 

54.  va^^xy/;:^;. 

65.  Va\/a2  x  V^/rT. 

56.  ■N/.r  V-^  X  V'j^^^. 

57.  -TxV  X  V-c^- 
_-  */a62        «/r.r5 


59.  V2.S  -^  ■V7.      • 

60.  Vis  ^  V^. 

61.  V;52  -  V2. 

62.  </ri{\  --  ^7. 

63.  V243  -^  73. 

64.  \/l2  ^  VO- 

65.  V54  ^  ^30. 

66.  v^  -=-  -y^- 

67.  Vrt^  ^  '\/2~a3. 

68.  \/«^P  -^  -V"^^- 

69.  \/27-i29  -V-  V<i2^ 

70.  -{/Sxh/i  ^  2  x2//3. 


Express  with  fractional  exponents  instead  of  radicals: 


71. 

i</m^y. 

72. 

(V^)^ 

73. 

(7^)^ 

74. 

(y^^y. 

75. 

i-^^y. 

Rationalize  the  denominator  of 

1 

81. 

V2' 

82. 

a 
a 

83. 

w 

84. 

(I 

1  +  Va 

85. 

1  -  sja 

86. 

^a  +  ^/b 
V«  -  Vb 

76.  (-  V^)^ 

77.  (7 V^. 

78.  (Vvil^)'' 

79.  (</{F+^3y 

80.  (\/  V-^'"rO'. 


87. 
88. 


90. 

91. 


3  +2  Va3 

2  Vo^  -  3 

1 


V-r  +  z/  -  V 

V8  +  V7 
V7  -  V2' 

VT3  -  yio. 

V 10  + Vis 
1 ; 

fl  '\/b  +  c  '\/d 


92.    Calculate  to  three  decimal  places  the  values  of  the  fractions  in  exer- 
cises 89  and  90. 


103.  (m-^  +  m-^y. 

104.  ia~^x-^ -ax^y. 

105.  {a'^+a^-J)\ 


26  EXERCISES  [33 

Perform  the  following  operations  and  simplify  results: 

93.  Va  <Jca  X  '^{/a^  X  ^/a2  </a7. 

94.  {x-^+x-iy~^+y~')ix~^  -x-^2/-=  +2/-^. 

95.  (2o-*-3a-^  +  a-*  -2)  (a-*  -2a-^ +3). 

96.  Write  out  the  result  of  replacing  a~^  by  6  in  exercise  95. 

97.  (a^  -  2  a^  +  3  a^J  \2  a^  -  a^  +  2). 

98.  {yn  -  ayn  +3  hyn  -  c)  \yn  +  byn  -  cyV. 

99.  (2a-i6-'  -3a-h-"-y. 

100.  (a- ^+6-^)1 

101.  (xi-y^y. 

102.  (l-'-n-^y. 

106.  (2a^  -36'^-4c^)^ 

107.  (a*  -2  6*  +3  c^  -iSy. 

108.  (.r*  ?/'  -  2  x'  y^  +  3  x^  2/  -  2  a;^  2/^) ^ 

109.  Write  out  the  result  of  replacing  x^  by  u  and  y*  by  f  in  exercise  108, 

110.  (x-l)  ^  {</x  -  l). 

111.  (x  +  1)  -(V^  +  l).  ^ 

112.  (Vx  -  V^)  -  (V5  -  -Ty)- 

113.  (a«  -6^)  ^  (a"-6''''). 

114.  (.r=  -  xy^  +  x^y  -  y^)  ■^(\/~x  -  \/ij). 

115.  (a^  -  a^  -  4  a^  +  6  n  -  2  Va)  ^  (a^  -  4  V^  +  2). 

116.  (a^  -  &4  -  c*  +  2  \/6c)  (a^  +  6'  -  c*'). 

Express  the  following  in  the  form  a\/  —  1. 

117.  V^^;   V-25;    V-81- 

118.  V^^';   V^^2;   V-a;2n. 

119.  V^^81.  121.    7^^'256. 

120.  V^^-  122.   '<J-a20, 
123.  V^^25  -  V^^49  +  V-  121. 


124.  V-  a"  +  V-  a2  -  V-  4  a*. 

125.  V-  (wi  +n)2  +  V-('«  -^)^  -  V' 


33]  EXERCISES  27 

Multiply  and  reduce  to  the  form  a  +  b  yf—  1  :  (i=  y/  —  l). 
126.    (a+6  V^^)(a-bV^)-     129.    {s/S  +  i  y/r2)  {y/2  +  i  yj3  ) . 

130.    (-  1  +;:  V3)^ 


127.  (3+5i)(4- 

128.  ix  +  2i)iy  - 

7i). 
Si). 

-(-^-■^r 

Reduce  to  the  form 

I  a  +  bi  by  rationalizing  the  denominator 

135.    -  +  '^- 

a  —  i  yx 

133.    '+'. 

136             ^^ 

1  -  i 

7+2  V-5 

134.   <■  +  «. 

a  —  01 

-  <1^- 

Clear  the  following  equations  of  radicals: 

(Example.     To  clear  the  equation  V^  +  V^  +  Vz  =  1  of  radicals  put 
Vx  +\/y  =  1  -  V2; 
squaring,  x  +  y  +  2  ^Jxy  =  I  +  2  -  2  y/z, 

or,  x  +  y— z  —  l=r-2  \/xy  -  2  yfz. 

Squaring  again,  (x  +  y  -  z  -  1)^  =  4  xy  +  iz  +  8  \lxyz, 

or  (x  +  ?/  -  2  -  1)2  -  4  (xy  +  2)  =  8  V^- 

Squaring  again,  [(x  +  y  —  2  —  1)2  —  4  {xy  +  2)]2  =  64  xyz.     q.e.f. 

138.  \/^T4  =  4.  145.    VxT20  -  V^^^  -3  =  0. 

139.  V2F+6  =  3.  146.  Vx        ^  '        " 


Vx 


140.   V-';  +  1  =  2. 


147.  Vl5  +  V2a;  +  80  =  5. 

141.  V-  +  6  =  c.  ^^g^  ^g-^-^  ^  VlT^iVi. 

142.  Vx  +  Vy  =  1^ ,,9_  V.^TVx  =  -^^^^^x. 

143.  V-r  +  1  -   V^;  -  1  =  2.  ,- —    ,      ,— 

150.  V8^±1±V|^=13. 

144.  V32+X  =  16  -  Vi.  VSx  +  l  -  VS X 


CHAPTER  III 

Logarithms.     Binomial  Theorem  for  Positive  Integral 
Exponents  / 

34.  Logarithm.  —  The  simple  laws  of  operation  for  exponents 
have  given  rise  to  a  method  of  calculation  involving  the  use  of  a 
function  called  the  logarithm.     We  shall  first  illustrate  this  method. 

Suppose  that  we  know  the  powers  of  10  which  are  required  to 
produce  a  set  of  numbers,  as  in  the  adjacent  table,  where  the 
exponents  are  given  to  the  nearest  figure  in  table. 

the  third  decimal.     The  exponent  of  10  in  each  5.00  =  lO^-^s^ 

equation   is   called   the   common   logarithm   (or  5.50  =  lO'^-^**^ 

the  logarithm  to  the  base  10)  of  the  number  on  6.00  =  10°-^'^ 

the  left.     Thus,  the  logarithm  of  5.00  is  0.699,  6-50  =  lOO-^is 

,  .  ,•  7  nn  —  100-845 

of  5.50  IS  0.740,  and  so  on.      As  equations,  ^'^^  "  \qo.s75 

we  write  ^^^  ^  i^omz 

logio  5.00  =  0.699,  8  50  =  iQO-^^a 

logio  5.50  =  0.740,  9.00  =  W-^^^^ 

9.50  =  l00-9"8 
10.00  -  lO^ooo 

35.  By  aid  of  such  a  table  products  of  numbers  (within  certain 
limits)  can  be  obtained  by  adding  the  logarithms  of  the  factors; 
also,  division  is  reduced  to  subtraction  of  logarithms. 

Example  1.    Find  the  value  of  6.5  X  8.5  X  9.5. 

We  have  6.5  X  8.5  X  9.5  =  IQO-sisx  100-929  x  100-978 

=  100.813+0.929+0.078 

=   102.720   =  102  X  100-720. 

Now  0.720  lies  almost  exactly  midway  between  0.699  and  0.740;  hence  the 
number  corresponding  to  100-720  ^ill  be  midway  between  5.00  and  5.50  and  is 
equal  to  5.25.  (This  involves  the  assumption  tliat  a  logarithm  changes  pro- 
portionately to  the  change  in  the  number,  an  assumption  which  is  not  exactly, 
but  very  nearly,  true  except  for  numbers  near  zero,  provided  the  changes  in 
the  numbers  are  small.) 
Therefore, 

6.5  X  8.5  X  9.5  =  102  X  100-720  =  lOO  X  5.25  =  525. 

The  exact  value  is  524.875. 

28 


and  so  on. 


36]  LOGARITHMS  29 

Definition.   Interpolation  is  the  process  of  calculating  numbers 
intermediate  to  tliose  given  in  a  table. 


Example  2.  - 

T..    ,  .,         ,        ,  6.25  X  7.20 
-  Find  the  value  of r^:          • 

Let 

10'^  =  6.25;  10''  =  7.20;  10^  =  5.75. 

Then 

6.25X7.20       lO-^XlO''       ^^^,_ 

5.75  lO*; 

Since  6.25  lies  halfway  between  6.00  and  6.50,  we  take  for  a  the  value 
halfway  between  the  corresponding  exponents,  so  that  a  =  0.795  (more  exactly 
0.7955).  To  get  b,  we  note  that  7.20  lies  g  of  the  way  from  7.00  to  7.50;  hence 
we  take  for  b  the  number  lying  in  the  corresponding  position  between  the 
.exponents  0.84:5  and  0.875.     Therefore 

b  =  0.845  +  I  X  0.030  =  0.857. 
Similarly,  c  =  0.759. 

fi  "^^  V  7  90 
Hence  —  =  100.795+0.857-0.759  =  loo-sas. 

'  5.75 

The  corresponding  number  lies  between  7.50  and  8.00,  and  nearer  the  latter. 
Since  our  exponent,  0.893,  lies  H  of  the  way  from  0.875  to  0.903,  we  find  the 
number  lying  in  the  corresponding  position  between  7.50  and  8.00,  that  is, 
7.50  +  ^  8  X  0.50  =  7.50  +  0.32  =  7.82. 

Therefore,  ""  ' —  =  7.82  approximately. 

This  result  is  correct  to  two  decimals. 

36.  By  the  aid  of  our  table,  powers  and  roots  of  numbers  may- 
be found  by  applying  the  operations  of  multiplication  and  division, 
respectively  to  their  logarithms. 

Example.   Find  the  value  of  \/9.353. 

We  have  \/0^^=  (9.35)^. 

Let  9.35  =  10«;       then    (9.35)*  =  lO^''. 

From  the  table,  a  =  0.954  +  i\  X  0.024  =  0.971. 

Therefore,  -^9^3^  =  100-728  =  5.00  +  |f  x  0.50  =  5.35. 

A  more  accurate  value  is  5.335,  so  that  the  second  decimal  of  our  result  ia 
slightly  in  error. 

Obviously  the  calculation  of  the  last  result  by  the  methods  of 
arithmetic  would  be  very  tedious,  and  with  a  slight  increase  in 
the  complexity  of  the  exponent  these  methods  would  become  quite 
useless. 


30 


LOGARITHMS 


[37-39 


We  shall  now  consider  the  general  theory  of  the  method  illus- 
trated above. 

37.  Logarithm  of  a  Number.  —  Let  a  be  a  certain  fixed  number, 
n  any  other  number,  and  let  x  be  the  exponent  of  a  required  to 
produce  n.     Then  x  is  the  logarithm  of  n  to  the  base  a. 

As  equations, 

if  a^  =  n,  then   a?  =  loga  n. 

We  give  below  some  very  simple  tables  of  logarithms. 


Number. 

Logarithm 
Base  =  2. 

n. 

logio  n. 

71. 

logio  n. 

1 
\ 
h 
1 
2 
4 
8 

-  3 

-  2 

-  1 
0 
1 
2 
3 

.001 
.01 

.1 

1.0 
10 
100 
1000 

-3 

-  2 

-  1 
0 

1 

2 
3 

5.00 
5.50 
6.00 
6.50 
7.00 
7.50 
8.00 

0.699 
0.740 

0.778 
0.813 
0.845 
0.875 
0.903 

38.  Exercises. 

1.  What  is  the  value  of  logo  1? 

2.  What  are  the  logarithms  of  8,  16,  64,  128  to  the  base  2? 

3.  What  are  the  logarithms  of  8,  16,  64,  128  to  the  base  i? 

4.  What  are  the  logarithms  of  \,  aV,  2I3,  to  the  base  3?  to  the  base  i? 

5.  What  are  the  logarithms  of  i|^  and  W  to  the  base  i? 

6.  What  are  the  logarithms  of  2,  4,  8  to  the  base  64? 

7.  What  is  the  base,  if  log  2  =  1?  if  log  a  =  1? 

8.  What  is  the  base,  if  log  ^  =  4?  if  log  25  =  -  2? 

9.  What  is  the  base  if  log  49  =  2?  if  log  .0081  =  4? 

10.  Whatislog2(-4)? 

11.  Why  would  it  be  inconvenient  to  use  a  negative  number  as  the  base  of 
a  system  of  logarithms? 

12.  If  n  =  {e^+y)^-y,  find  loge  n. 

13.  If  X  =   ^e  {[eieP+i),  find  loge  x. 

I"  ,      ,    J^-^x'-xy+v^  •  '; 

14.  If  a  =  [  (10^     2^  )  ^-A  ,  find  logio  a. 

15.  Show  that  a'oga^  =  x. 

39.  Laws  of  Operation  with  Logarithms.  —  Since  a  logarithm  is 
an  exponent,  the  laws  of  operation  for  logarithms  are  the  same  as 
those  for  exponents.    ^ 


40]  LOGARITHMS  31 

Let  X  be  the  logarithm  of  m,  y  that  of  n,  the  base  being  a. 
Then 

I  logo  m  =  X,  {a'^=  m, 

I  loga  n  =y, 


Hence 


mn  =  a^  +  ^     and     — 
11 


or,  log„  mn  =  x  -{-  y  =  log„  tn  +  log„  n, 

and  loga  —  ^  X  —  y  =  log„  w»  —  log„  n. 

n 

We  have  therefore  the  rules: 

I.  The  logarithm  of  a  product  equals  the  sum  of  the  logarithms  of 
the  factors. 

II.  The  logarithm  of  a  fraction  equals  the  logarithm  of  the  numer- 
ator minus  the  logarithm  of  the  denominator. 

Also,  if  as  before, 

loga  fn  =  X,     so  that    m  =  a^, 
then,  if  p  and  q  be  any  real  numbers, 

X 

mP  =  a^'     and      •\/m  =  a^. 
Hence,  log„  m^'  =  px  =  p  log„  tn, 

X      1 

and  loga  Mm  =  "^=  ^  log„  rn. 

We  have  therefore  two  additional  rules: 

III.  The  logarithm  of  any  power  of  a  number  equals  the  ex- 
ponent of  the  power  times  the  logarithm  of  the  number. 

IV.  The  logarithm  of  any  root  of  a  number  equals  the  logarithm  of 
the  number  divided  by  the  index  of  the  root. 

(Rule  III  contains  rule  IV,  since  the  power  in  question  may  be 
fractional.) 

40.  The  following  facts  regarding  logarithms  should  also  be 
carefully  noted. 

(a)  In  any  system  the  logarithm  of  the  base  is  1. 

For  d^  =  a.  :.   loga  a  =  1. 

(6)  In  any  system  the  logarithm  of  1  is  0. 
For  ao  =  1.  .-.   loga  1  =  0. 


32  LOGARITHMS  [41 

(c)  In  any  system  whose  base  is  greater  than  unity,  the  log- 
arithm of  0  is  —  CO. 

For  if  d^  =  m  and  a  >  1,  then  if  a;  is  a  large  negative  number, 
m  will  be  small.  As  x  increases  indefinitely,  always  being  nega- 
tive, m  approaches  zero.     That  is, 

a-°==Oifa>l;  /,   log  0  ==  -  oo. 

(d)  A  negative  number  has  no  (real)  logarithm,  the  base  being 
positive. 

(e)  As  a  number  varies  from  0  to  -|-oo,  its  logarithm  varies 
from  —CO  to  +«3,  the  base  being  greater  than  1. 

Wlien  the  number  is  greater  than  1,  its  logarithm  is  positive. 
When  the  number  is  less  than  1,  its  logarithm  is  negative. 
41.    Exercises.      (See  Appendix  for  tables  and  explanation  of 
their  use.) 

1.  Discuss  (c)  of  (40)  when  the  base  is  less  than  unity. 

2.  Discuss  (e)  of  (40)  when  the  base  is  less  than  unity. 

In  the  following  exercises,  the  base  is  understood  to  be  10,  and  four-place 
logarithms  are  to  be  used. 

3.  Find  log  831,  log  8.31,  log  .831,  and  log  .0831. 

4.  Find  log  78.03,  log  .073,  log  .00284. 

5.  Find  the  approximate  value  of  564.1  X  .0065. 

6.  Calculate  \/ 154.2  and  (7.541)3. 

7.  Calculate  518  ^  313  and  25.03  ^  2.14. 

8.  Calculate  .001022  -;-  .0000513  X  1.415. 

9.  Calculate  17  V29  and  41  VO.512. 

10.  Calculate  7o^*  X  <JQA7'^. 

,,     „  ,     ,  ,    (.00165)3  (.07(>4)2 

11.  Calculate -^3^35^^, 

12.  Calculate  ^214  -  V2li. 


Write  as  a 

single  term : 

13. 

log  a 

-  log  6  +  log 

c  -  log  d. 

14. 

Slog 

X  -  4  log  2/  + 

2  log  z. 

15. 

ilog 

M  +  J  log  y  - 

\  log  w. 

16. 

log^ 

+  log  ~  +  log 

c       ,      ax 
d-^^'^dy 

J  log  {ax  +h)+  log  V«-c  +  b. 


42,43]  BINOMIAL  THEOREM  33 

The  Binomial  Theorem  for  Positive  Integral  Exponents 

42.  This  theorem  is  used  to  express  (a  +  6)"  in  expanded  form. 
We  shall  here  obtain  the  formula  assuming  n  to  be  a  positive 
integer;  the  proof  for  other  values  of  n  will  be  found  in  (221). 

By  actual  multiplication  we  have 

(a  +  6)2  =  a2  -H  2  ah  -\-  b^, 

(a  -I-  &)3  =  ^3  _f_  3  ^2^  _^  3  (iJ^2  ^  ^3^ 

(a  +  6)4  =  a4  +  4  a%  +  6  a262  +  4  ab''  +  6^. 

Here  we  observe  the  following  laws:  * 

I.  The  number  of  terms  is  1  greater  than  the  exponent  of  the 
binomial. 

II.  The  exponent  of  a  in  the  first  term  equals  that  of  the  bino- 
mial and  decreases  by  unity  in  each  succeeding  term.  The  ex- 
ponent 0/  6  is  1  in  the  second  term  and  increases  by  unity  in  each 
succeeding  term. 

III.  The  coefficient  of  the  first  term  is  1,  and  of  the  second 
term  the  exponent  of  the  binomial.  If  the  coefficient  of  any 
term  be  multiplied  by  the  exponent  of  a  in  that  term,  and  the 
result  be  divided  by  the  exponent  of  b  plus  1,  we  obtain  the 
coefficient  of  the  next  following  term. 

43.  Now  let 

(1)  (a+6)"  =  a«+cia'^-i6+C2a"-262+  •  •  •  +c,„_  ia"-^'"-i)6'"-i 

+c,„a''-'"6'"+c,„+ia"-('»  +  i)6™  +  i+  •  •  •  . 

We  have  here  assumed  laws  I  and  II  and  have  written  the  ex- 
ponents accordingly.     Assuming  also  law  III,  we  shall  have 

._,,  n  —  1  w  —  (w  —  1)  n  —  m 

(2)  ci  =  n;  C2  =  ^ 2"  ci ;  c,„  = Cm-  1 ;  c,„+  1  =  ^— t^c,^. 

We  can  now  show  that  the  same  laws  are  true  for  the  expan- 
sion of  (a  +  bY'^^. 

Multiplying  (1)  by  (a  -|-  b)  and  collecting  like  terms  we  have 

(3)  (a+6)"  +  i=a"  +  i  +  (l+ci)a("  +  i)-i6+(ci+C2)a("  +  i>-262H 

+  (c,„_i+c,„)a"  +  i-"'6'"  +  (c,„+c,„  +  i)a("  +  i)-("*  +  i)6'«  +  i+--- 

The  number  of  terms  will  be  n  +  2,  since  the  exponent  of  a  starts 
with  n  +  1  and  decreases  to  0.  Hence  law  I  is  still  true.  Also 
law  II  is  evidently  true. 


34  BINOMIAL  THEOREM  [44 

According  to  the  third  law,  we  should  have 

(l+ci)=n  +  l;  ci+C2-^''"^2^~ -^(l+ci);  •  .  . 

{Cm  +  Cm+i)   =  ^i-l-l ^^^-  1   +  ^"'^^ 

These  equations  all  become  identities  on  substituting  from  (2). 

Therefore  all  three  laws  are  true  for  the  expansion  of  (a  +  6)"+^ 
provided  that  they  are  true  for  the  expansion  of  (a  +  6)".  But  they 
are  true  for  (a  +  6)^,  hence  for  (a  +  b)-^,  hence  for  (a  +  b)^,  and  so 
on,  for  any  positive  integral  exponent. 

This  method  of  proof  is  called  proof  by  induction. 

Writing  out  the  values  of  several  coefficients  we  have, 

n  (n  -  1)             n  (n  -  1)  (w  -  2) 
ci=n;  C2=      ^  ^  2      '  ^^  = TTTs '  *  *  * 

_  n  (n  -  1)  (n  -  2)  .  .  .  (n  -  ?n  +  1) 
^'"~  1  .  2  .  3  .  .  .  .  w 

where  c^  is  the  coefficient  of  the  (m  +  l)th  term. 

In  place  of  1  •  2  •  3  •  .  .  .  w  we  use  the  symbol  [w  or  m!  (in 
either  case,  read  "factorial  m").     Then  equation  (1)  becomes 

,  ,  7i(n-l)(n-2)  .   •   •   (u-m  +  1)    ,._„,,„,, 

I'm 

When  o  =  1  and  &  =  a;  we  have, 

\2  [3 

44.  The  expansion  of  (a  +  b)"  may  be  reduced  to  that  of 
(1  +  x)"  thus: 

(a  +  br  =  a"(l  +  ^y  =  fl4l  +  'i^  +  •  •  •]• 

In  place  of  Cm  to  denote  the  coefficient  of  the  (m  +  l)th  term 
of  the  expansion  of  (a  +  6) ,  the  symbols  nCm  or  {^J  are  often 
used.     These  are  called  the  binomial  coefficients. 


45]  BINOMIAL  THEOREM  35 

Table  of  Binomial  Coefficients 


w  =  0 

1 

n  =  1 

1     1 

n  =  2 

12       1 

71  =  3 

13      3      1 

n  =  4 

1 

4      6       4     1 

n  =  5 

1 

5    10     10      5     1 

Example  1. 

Expand  (a*  -2  6^)*. 

[{ai)+{-2lr)Y 

=  (a*)' 

'+4C 

a^n-2l/)+Q(a^)H-2l^r 

+  4C 

ai)(-2by  +  (-2b'y 

=  a'- 

8a¥ 

+  24  a6^  -  32  aH^  +  16  6^ 

Example  2. 
Find  the  fifth  term  in  the  expansion 

of(x-i-hy'r. 

This  term  will  be 

8-7-6 
1  .2-3 

^(- 

^K- 

-^^r=f-^^^- 

45.   Exercises.     Expand: 

1.  ix-y)^. 

2.  (2a -3  6)6. 

3.  (a-i  +6-2)4. 

4.  (x-i-rr. 
6.  (x^+rr. 

12.  (1  +  a^)7. 

13.  (a^  +  62/)6. 

14.  (a^+2/+a^-2/)5. 

15.  (x«-'  -  2/"»')6. 

6.    {2p--3q'-y. 

16.    {x'  -  yyy. 

7.    (ax  +  6!/)8. 

17.    (^/'.x  +  ^yY. 

8.    (^u-2+2!;2)7. 

18.    (x^^«  -«^^)^ 

9.    {^2~x  -  </3ijy. 

19.     (e2.r+xc-2^)5. 

-(^-^y- 

\  n2       m2/ 

To  expand  a  trinomial  or  other  polynomial,  proceed  by  grouping  the  terms 
in  two  groups,  thus: 

(x  +  2/  +  2)3  =  [x  +  (y  +  zW 

=  x3  +  3  x2  (y  +  2)  +  3  X  (y  +  2)2  +  (;/  +  2)3. 

The  expansion  may  now  be  completed  by  the  formula. 

21.  {x-\-y  -  2)3.  24.    {x-y  +  u  -  v)^. 

22.  (Vx-Vi/  +  V2)^  25.    (l+2x +3x2 +  4x3)3. 

23.  (1  +  2  a  +  3  a2)4. 


36  BINOMIAL  THEOREM  146 

Calculate: 

26.  the  6th  term  of  (3  +  2  x^)^. 

27.  the  5th  term  of  {'\/2~c  +  V37z)io. 

28.  the  Sth  term  of  (2  6^  -  i  V^)^- 

29.  the  12th  term  of  (3  i/  +  J  y^Y^. 

30.  the  10th  term  of  (Vl~^3  _  Vl^)™. 

16.   Approximate  Computation  by  Use  of  the  Binomial  Theorem. 

—  When  re  is  a  small  fraction,  the  terms  of  the  formula 

rapidly  decrease.  In  amj  numerical  problem  in  which  only  approxi- 
mate results  are  required,  retain  only  enough  terms  of  the  expansion 
to  obtain  the  desired  degree  of  accuracy. 

It  will  often  be  found  sufficient  to  use  the  simple  formula, 

(1  +  a^)"  =  1  +  nx,  approximately. 
Example  1.     Calculate  (0.997)4  to  three  decimals. 

(0.997)4  =  (1  -  .003)4  =  1  -  4  X  .003  =  0.988. 

Exercise.     Show  that  the  terms  neglected  will  not  affect  the  third  decimal 
place. 

Example  2.     Calculate  (2.05)3  to  three  decimals. 

(2.05)3  =  23  (1  +  .025)3  =  8  (1  +  3  X  .025  +  3  X  .000625  +  •  •  •) 
=  8  X  1.0769  =  8.615. 

Exercises.     Calculate  to  three  decimal  places  the  value  of: 

1.    (0.995)5.  2.    (1.05)7.  3.    (3J,)4. 

4.    (2Uy.  5.    (3.998)6.  6.    (8.0125)2. 

7.    Calculate  the  value  of  (.99995)7  to  seven  decimals. 


CHAPTER   IV 

Linear  Equations 

47.  If  X  =  Y,      and    7n  =  n, 

then  X  +  ?n  =  F  -f  n,       X  —  m  =  Y  —  n, 

mX  =  nY,     and    —  A'  =  -  Y. 

m  n 

That  is,  if  both  members  of  an  equation  be  increased  or  diminished, 
multiplied  or  divided,  by  the  same  or  equal  quantities,  the  results 
are  equal. 

Also  if  X  =Y,  then  X'*  =  F", 

n  being  an  integer;  that  is,  if  both  members  of  an  equation  be  raised 
to  the  same  iyitegral  power,  positive  or  negative,  the  results  are  equal. 

If  A'  =  F,    then    VaT  =  VF, 

provided  the  corresponding  nth  roots  of  X  and  F  are  selected. 

If  X-\-m=  Y, 

then  subtracting  m  from  both  members, 

X  =Y-m. 

That  is,  a  tertJi  may  be  transposed  from  one  side  of  an  eqiiation  to 
the  other  provided  its  sign  is  changed  at  the  same  time. 

When  the  members  of  an  equation  involve  sums  or  differences 
of  fractions,  the  equation  may  be  cleared  of  fractions  by  multiply- 
ing both  members  by  the  L.  C.  D.  of  the  several  fractions. 

48.  Linear  Equation.  —  If  a:  be  an  unknown  quantity  related 
to  the  known  quantities  a  and  b  through  the  equality  ax  -\-  h  =  0, 
this  equation  being  called  the  standard  form  of  the  linear  equation 
in  one  unknown,  we  obtain  the  value  of  x  as 

b 

X  = 

a 

37 


38  LINEAR  EQUATION  [49,50 

Every  linear  equation  in  one  unknown  may  he  solved  by  reducing 
it  to  standard  form  and  applying  the  last  formula. 

The  reduction  of  an  equation  to  standard  form  will  involve 
some  or  all  of  the  following  steps: 

1.  Clearing  of  radicals.     (33,  after  exercise  137.) 

2.  Clearing  of  fractions. 

3.  Expanding  products  or  powers  of  polynomials. 

4.  Transposing  and  cancelling. 

5.  Collecting  terms. 

To  verify  the  value  found,  substitute  it  in  the  given  equation. 
The  result  should  be  an  identity. 

49.   Example  1.     Solve  for  x:    {I  +  b)  x  +  ab  =  b  (a  +  x)+  a. 
Expanding  the  products:  x+i^+(^  =  aJf+Mx  +  a. 

Cancelling  like  terms:  x  =  a 

Check:  (1  +  b)  a  +  ab  =  b  {a  +  a)  +  a. 


Example  2. 

•       1,2          X  +2 
2      .T  +  2         2x 

Multiplying  by 

the  L.  CD.,  2  x{x  +2): 

x(x+2)  +4x  =  {x  +  2)2. 

Expanding: 

x2  +  2  a;  +  4  X  =  .r2  +  4  x  +  4. 

Cancelling: 

2  X  =  4     or     x  =  2. 

Check: 

§  +  l  =|. 

Example  3. 

Solve  for  x:      \/x +  20 -^x  -  1  -  3  =  0. 

Transposing: 

Vx  +  20  =  V-c  -  1  +  3. 

Squaring: 

X  +  20  =  X-  -  1  +  6  V.r  -  1  +  9, 

or, 

2  =  V-c  -  1- 

Squaring: 

4=x-l     or     x  =  5. 

Check: 

V27j  -  V4  -  3  =  0. 

50.   Infinite  Solutions.  —  Consider  the  equation 


X  -\-  1      X  —  1 

Since  x  -\-  1  cannot  equal  x  —  1  for  any  value  of  x,  there  is  no 
value  of  X  which  will  satisfy  the  given  equation. 

But  if  we  substitute  in  the  given  equation  successively  x  =  10, 
100,  1000,  etc.,  the  equation  is  more  nearly  satisfied,  the  larger 
the  value  of  x.     We  can  take  x  so  large  as  to  make  the  differ- 


51,52]  LINEAR  EQUATION  39 

.ence  between  the  two  members  of  tlie  equation  as  small  as  we 
please;  for  this  difference  is 

_l 1      ^     -2 

rc+1       X  —  I       .T^  —  1 

For  brevity  we  say  that  x  =  <^  is  a  solution  of  the  equation, 
meaning  thereby  that  as  increasing  values  of  x  are  substituted, 
the  equation  is  more  and  more  nearly  satisfied. 

Substituting  formally  x  =  oo,  we  obtain 

^  ^    or   0  =  0. 


00  +  1  c 

The  equation  of  example  2  of  (49)  admits  the  solution  a;  =  ^. 
This  will  be  evident  on  putting  cc  for  x  in 

2'^a;  +  2         2x        2"^  re* 

51.   Exercises.     Solve  for  x,  including  infinite  solutions  when 
present : 

1.  5  (a  -x)  =3(6  -x). 

2.  pix  -I)  +x  =  q  -  p. 

3.  a  (bx  —  e)  =  ae  —  abx. 
.  m  —  X  _x  —  n 


6. 

7n   ,           p   , 

6. 

a  +  bx      a 
c  +  dx      c 

7. 

a  +  bx      b 

c+dx      d 

8. 

a  +  b        e  -  d 
a  +  bx      c  —  dx 

a  +  l 

9. 

X          a  +x 
b          b+x' 

10. 

ex  +  d 

m          2d 
ex           VI 

d 

11. 

m              n           in  —  n 

X  —  VI       X  —  n       X  —  a 

12. 

Vx  +  15  +  V-c  -  13  =  14. 

13. 

3  Vl6x  +  9  +9  =  12V4X. 

14. 

Vx  -  V^  -  5  -  V5  =  0. 

15. 

y/T+x  =  l  +  ^/x. 

16. 

1        *>       3 

52.    Graphic  Solution  of  Linear  Equations.  —  Suppose  that  a 
given  equation  has  been  reduced  to  the  standard  form, 


40  GRAPH  OF  LINEAR  EQUATION  [53 

The  solution  of  the  equation  is  that  value  of  x  which  reduces  the 
binomial  to  0.  For  brevity,  let  us  represent  the  binomial  by  y, 
so  that 

y  =  aoc  +  b. 

Then  we  want  that  value  of  x  for  which  y  =  0.  If  now  we  form 
a  table  which  gives  the  values  of  y  corresponding  to  a  series  of 
assumed  values  of  x,  we  may  obtain  from  it  by  inspection  the 
exact  or  approximate  value  of  x  for  which  y  is  zero. 

Example. 

Let  2  a;  -  1  =  0  so  that  y  =  2  x  -  1. 

Corresponding  values  of  x  and  y  are: 

X  =-2,     -  1,         0,     +1,     +2,     +  3,  .  .  .  . 
y  =-  5,     -  3,     -  1,     +1,     +3,     +  5,  .  .  .  . 

By  inspection  we  see  that  y  =  0  when  x  Ues  between  0  and  1. 

53.  Graph  of  the  Equation  y  =  2  x  —  l.  —  We  shall  now  repre- 
sent the  corresponding  values  of  x  and  y  graphically. 

Divide  the  plane  of  the  paper  into  four  quarters  or  quadrants 
by  drawing  two  mutually  perpendicular  lines,  XX  and  YY, 
intersecting  at  0.  (See  figure.) 
Adopting  any  convenient  unit  of 
,,  ,t  ,j  length  (say  one-fourth  of  an  inch,  or 
~'  ''^^'  '  one  side  of  a  square  of  the  cross- 
section  paper),  mark  on  XX  a  series 
of  points  whose  distances  from  0 
shall  equal  the  assumed  values  of  x.  When  x  is  positive,  the 
distance  is  laid  off  to  the  right  from  0;  when  x  is  negative,  to 
the  left. 

In  this  way  all  positive  and  negative  integral  values  of  x  are 
represented  by  a  series  of  segments  having  a  common  starting 
point  0,  and  ending  in  a  series  of  equally  spaced  points  on  the 
line  XX,  each  of  which  represents  an  integral  value  of  x.  Non- 
integral  values  of  x  are  represented  by  segments  whose  end  points 
fall  between  two  points  representing  integral  values.  Thus  in 
the  figure  are  marked  the  points  corresponding  to  .r  =  ±1,  ±2, 
±3,  H-2^  and  -If. 

Now  to  represent  the  value  of  y  corresponding  to  a  given  value 
of  X,  mark  the  representative  point  of  x  on  A^Y,  and  at  this  point 
lay  off  a  segment  perpendicular  to  XX  and  having  a  length  equal 


T)!] 


GRAPH  OF  LINEAR  EQUATION 


41 


>■   ■ 

n 

p^ 
y- 

3 

r, 

fl 

Y 

to  the  value  of  y;  this  segment  is  drawn  upward  when  y  is  posi- 
tive, and  downward  when  y  is  negative. 

When  we  construct  in  this  way  the  pairs  of  values  of  x  and  y 
given  in  the  example  of  (52),  we  obtain  the  figure  below.  We 
thus  get  a  series  of  points.  Pi,  P2,  .  .  .  ,  Pg,  whose  distances 
from  the  line  XX  are  the  values  of  the  binomial  2  a;  —  1  for  the 
assumed  values  of  x.  Inspection  of  the  figure  shows  that  as  x 
increases  from  —2  to  +3,  y  (i.e.  2  a:  —  1) 
increases  from  —5  to  +5;  also  that  y  =  0 
between  a:  =  0  and  1. 

Exercise.  By  similar  triangles,  show  that  any 
three,  and  hence  all  the  points  marked  in  the  figure, 
lie  on  a  straight  line. 

By  drawing  a  smooth  curve  (in  this  case  a 
straight  line)  through  a  sufficient  number  of 
points  Pi,  P2,  ...  we  obtain  the  gra-ph  of 
the  equation  y  =  2  x  —  1.  The  points  Pi, 
P2  .  .  .  are  said  to  lie  on  this  graph. 

54.  Graph  of  »/  =  ax  +  b. — The  graph  of  the  equation  y  =ax-\-b 
is  a  geometric  picture  which  indicates  the  value  of  y  correspond- 
ing to  any  assumed  value  of  x. 

We  shall  now  show  that  this  graph  is  a  straight  line,  by  show- 
ing that  any  three  of  its  points  are  collinear. 

Let  xi,  X2,  and  xs  be  any  three  values  of  x;  let  yi,  2/2,  and  2/3 
be  the  corresponding  values  of   y.     Lay  off   the  corresponding 
and  (xs,  2/3)  so  that  (see  figure) 
X,  =OMu  yy  =MiPi, 

2/2   =  M2P2, 

X3  ■■ 

But  since 

by    putting    x  =  Xi    in    y  =  ax  -\-  b,    and 

similarly  for  ?/2  and  7/3,  we  have 

axi-{-b 

2/2  -  2/1  =  a{x2  -  X,), 

ys  -  2/2  =  a(x3-  X2). 


values  (xi,  yi),  {xo,  2/2) 
P, 


K 


OM2, 

OM3,  1J3   =  M3P3. 

7/1   is   the   value   of  y   obtained 


M,   Af>   M, 


7/2  =  ax2  +  b 
ys  =  axs-\-b 


Therefore, 


2/2  -  .Vl    ^  2/3  -  ?/2 
X2 


Xi 


X3  -  X2 


i=a). 


42  GRAPH  OF  LINEAR  EQUATION  [55,56 

But       y2  -  yi  =  M2P2  -  Ml  Pi  =  M.Po  -  M.H  =  HP2; 
?/3  -  1/2  =  il/sPs  -  M2P2  =  KPs; 
X2-X1  =  OM2  -  OMx  =  M1M2  =  PiH; 
and        0:3  -  3:2  =  OM3  -  OM2  =  M2M3  =  P2K. 

Substituting  these  in  the  two  fractions  above  we  obtain 
HP2^KPs 
PiH      P2K 

Therefore  A  P1HP2  is  similar  to  A  P.KP^. 
Hence  the  points  P\ ,  P2,  P3  he  on  a  straight  hne. 

Theorem:  The  graph  of  the  equation  y  ^  ax  -{-b  is  a  straight  line. 

Corollary:   To  construct  the  graph  of  the  equation  y  =  ax  -\-  h, 
construct  two  points  on  it  and  draw  a  straight  line  through  them. 

55.  Exercises.     Draw  the  graphs  of  the  equations  (each  set 
to  the  same  reference  Hues): 

1.  y  =  X  +  1,  2  2/  =  2  .T  +  2,  5  X  =  5  X  +  5,  -^  2/  =  i  X  +  |. 

2.  2/  =  3  X  -  4,  2  ?/  =  6  X  -  8,  %  =  3  Lc  -  4  /t. 

3.  2/  =  x  +  l,  2/=x+2,  ?/=x+3,  ?/=x-l. 

4.  2/  =  3  X  -  4,  2/  =  3  X  -  2,  7/  =  3  X,  2/  =  3  X  +  1. 

5.  2/=:c  +  l,  2/  =  2x  +  l,  2/  =  3x  +  l,  2/  =  ix  4jr*. 

6.  2/=3x-4,  2/  =  6x-4,  2/  =  9x-4,  2/  =  ix  --4. 

7.  2/=-^  +  l,2/=-3x-4. 

8.  2/  =  -c  -  1,  2/  =  3x  +  4. 

Explain  the  effect  on  the  graph  of  2/  =  ax  +  6,  of : 

9.  Multiplying  the  equation  through  by  a  constant. 

10.  Changing  the  value  of  h. 

11.  Changing  the  value  of  a. 

12.  Changing  the  sign  of  h. 

13.  Changing  the  sign  of  a. 

56.  Coordinates.  —  Divide  the  plane  into  four  quadrants  by 

the  lines  XX  and  YY  as  before,  and  let  P  be  any  point  in  the 

II  ,  I  plane  (see  figure),  obtained  by  laying  off  a 
[/'  pair  of  corresponding  values  of  x  and  y. 
\}i         The  position  of  P  is  completely  determined 

^  ^  I  ^ — \ — ^   as   soon  as  x  and  y  are  given.     Therefore 

P^  x  and  y  are  called  the  coordinates   of   P, 

X   being    called    the    abscissa,    and   y    the 

III  IV     ordinate. 

A  point  whose  coordinates  are  x  and  y  is  referred  to  as  the  point 


57]  .    GRAPH  OF  LINEAR  EQUATION  43 

The  four  quadrants  of  the  plane  are  numbered  consecutively 
as  in  the  figure,  and  are  called  the  first  quadrant,  the  second  quad- 
rant, and  so  on. 

The  line  XX  is  called  the  axis  of  x,  and  YY  the  axis  of  y. 
It  is  evident  (definitions  of  x  and  y  in  (53))  that  the  signs  of 
the  coordinates  in  the  four  quadrants  will  be  as  in  the  following 
table: 

Quadrant  Abscissa  Ordinate 

I  +  + 

II  -  + 

III 
IV  +  - 

57.  Linear  Equation  in  Two  Variables.  —  If  x  and  y  are  unre- 
stricted, the  point  {x,  y)  may  have  any  position  in  the  plane.  But 
when  a  relation  between  x  and  y  is  given,  as  y  =  2  x  or  y  =  x  -j-  1,  or 
2  X  —  3  y  -\-  4:  =  0,  the  point  (x,  y)  is  thereby  restricted  to  a  defi- 
nite path,  which  we  have  already  called  the  graph  of  the  equation. 

A  relation  of  the  form  Ax  +  Bij  +  C  =  0  is  called  the  general 
linear  equation  in  two  variables. 

Theorem:  The  graph  of  the  linear  equation  Ax  -\-  By  -\-  C  —  0  is 

a  straight  line. 

■  j^         (J 
Proof:  If  5  7^  0,  we  can  write  ?/  =  —  77  a:  —  75 ,  which  has  the 

t>         li 

form  y  =  ax  -\-b.     Therefore  the  graph  is  a  straight  line  when 

5  7^  0. 

(7 
If  5  =  0,  the  equation  reduces  to  Ax  -{-  C  =  0,  or  x  =  — ^' 

unless  A  =  0.    But  if  A  =  0  and  B  =  Q,  then  C  =  0  and  the  equation 

vanishes  identically.     Excluding  this,  we  may  reduce  Ax  -\-  C  =  0 

C 
toa:=  —  -j)Ora:  =  a  constant.    But  this  is  a  straight  line  parallel 

to  the  7/-axis.     Therefore  the  given  linear  equation  represents  a 
straight  line.     (Hence  the  term  "  linear  "  equation.) 

Exercises. 

4         C 

1.  Show  that  the  equations  Ax  -\-  By  +  C  =  0  and  y  =  —  '.,x  —  -  have 

the  same  graph. 

2.  Show  that  the  equations  Ax  +  By  +  C  =  0  and  kAx  +  kBy  +  kC  =  0 
have  the  same  graph,  k  being  any  constant. 

3.  How  is  the  graph  oi  Ax  -{■  By  +  C  =  0  aff(;ctcd  by  a  cliangc  in  C? 
inBfm  Af 


44 


USE  OF  THE  GRAPH 


[58 


58.  Use  of  the  Graph.  —  When  any  two  variable  quantities  are 
connected  by  a  linear  equation,  the  relation  between  them  can 
always  be  represented  graphically  by  a  straight  line.  It  is  only 
necessary  to  consider  the  two  quantities  as  the  coordinates  of  a 
point. 

Example  1.  A  man  starts  5  miles  south  of  A  and  walks  due  north  at  the 
rate  of  3  miles  an  hour.     How  far  is  he  from  A  at  the  end  of  x  hours? 

Solution.   Let   y   be  the  required  distance.      Also 
let  y  be  negative  to  the  south  of  A,  positive  to  the 
north.     Then  the  relation  between  y  and  x  is 
3x  -5. 


y 


Sx 


The  graph  is  shown  in  the  figure.  Here  one  square 
on  the  horizontal  scale  represents  one  hour,  and  one 
square  on  the  vertical  scale  represents  one  mile. 

Exercise.  By  inspection  of  the  graph,  find  the  dis- 
tance from  A  at  the  end  of  0,  2,  3,  4^  hours  respec- 
tively. Compare  with  the  values  obtained  from  the 
equation. 

In  this  example  negative  values  of  x  and  the  corre- 
sponding values  of  y  may  be  interpreted  as  follows: 
Let  the  time  be  counted  from  the  moment  when  the 
man,  supposed  to  be  walking  due  north  continuously 
at  the  rate  of  3  miles  an  hour,  arrives  at  the  point  5 
miles  south  of  A.  Let  time  after  this  moment  be 
called  positive,  and  before  it,  negative.     Thus,  3  hours 

before  this  moment  would  be  represented  by  x  =  —  3.     The  corresponding 

value  of  2/  is   —  14,  that  is,  the  man  was 

14  miles  south  of  A. 

Example  2.  The  relation  between  the 

readings  on   the   scales   of   a  Centigrade 

and  a  Fahrenheit  thermometer  is  given  by 

the  equation 

C  =  UF  -  32). 

Draw  the  graph. 

We  shall  retain  the  letters  F   and   C 
instead  of   replacing  them  by  x   and  y. 

The  graph  is  shown  in  the  adjacent  figure.  From  it  the  reading  of  either 
scale  corresponding  to  a  given  reading  of  the  other  may  be  at  once  read  off, 
with  an  accuracy  of  about  1°. 

Exercise.  Read  ofT  the  values  of  C  corresponding  to  F  =  —  40°,  F  =  0°, 
F  =  57°  respectively;  also  the  values  of  F  when  C  =-  30°,  0°,  +  21°. 

Example  3.  A  volume  of  gas  expands  when  the  temperature  rises  and  con- 
tracts when  the  temperature  falls  according  to  the  law 

V    =   I'O  (1    +   273  0, 


59,60] 


EXERCISES  AND  PROBLEMS 


45 


w 

, 

ZOOcu^ 

^ 

^ 

lOOai-ft. 

-soo-'  ' 

-zoo' 

-lOO' 

o'    '   '   ' 

UOO" 

*200' 

where  vq  =  volume  at  temperature  0°, 

and  V  =  volume  at  temperature  t°. 

Represent  graphically  the  rela- 
tion between  volume  and  tem- 
perature for  a  quantity  of  gas 
whose  volume  at  0°  is  100  cu.  ft. 
Replacing  273  l:)y  its  approxi- 
mate value  .0037,  and  I'o  by  100, 
the  equation  becomes 

V  =  .37  <  +  100. 
The  graph  is  given  in  the  adjacent  figure. 

59.   Exercises. 

1.  From  the  figure,  read  off  the  volumes  corresponding  to  the  temperatures 
250°,  75°,  0°,  and  —  273° ;  also  the  temperature  corresponding  to  the  volumes 
150,  75,  and  20  cu.  ft.  respectively. 

2.  Construct  a  graphic  conversion  table  for  converting  yards  to  feet. 

3.  Construct  a  graph  showing  the  relation  between  the  circumference  and 
the  diameter  of  a  circle. 

4.  A  falling  body,  starting  with  an  initial  velocity  of  vo  ft.  per  second, 
acquires  in  t  seconds  a  velocity  given  by  t;  =  gt  -{-  vo,  in  which  g  =  32.2.  As- 
sume a  value  of  vo  and  draw  the  graph  of  the  equation. 

5.  Let  A  be  the  lateral  area  of  a  right  circular  cylinder  of  height  h  and  radius 
of  base  r.  Draw  the  graph  showing  the  relation  between  A  and  h  when  r  is 
fixed.  Also  draw  the  graph  showing  the  relation  between  A  and  r  when  h 
is  fixed. 

6.  Same  as  5,  except  that  cone  is  substituted  for  cylinder,  and  slant  height 
for  height. 

Solve  for  x  graphically: 
7.    8  +  X  =  15. 


8.  3  X  =  27. 

9.  2  (x  -  1) 
10.    ix-FJx 

60.   Problems. 


11. 


12. 


x-2 


3x-5 


?x-l 


__  _8 
^x      3' 


1.  If  12  be  added  to  7  times  a  certain  number  the  sum  is  54.     What  is 
the  number? 

2.  Find  a  number  such  that  if  16  be  subtracted  from  it  and  the  result 
multiplied  by  5,  the  product  equals  the  number. 

3.  Find  a  number  such  that  if  a  be  subtracted  from  it  and  the  result  mul- 
tiplied by  m,  the  product  equals  the  number. 

4.  Find  a  number  such  that  3  times  the  number  increased  by  10  equals  5 
times  the  number. 

5.  Find  a  number  such  that  m  times  the  number  increased  by  a  equals  n 
times  the  number. 


46  SIMULTANEOUS  LINEAR  EQUATIONS  [61 

6.  The  age  of  a  boy  is  three  times  that  of  his  brother,  and  their  combined 
ages  make  16  years.     How  old  is  each? 

7.  In  what  proportion  must  two  Hquids,  of  specific  gravities  1.20  and  1.40 
respectively,  be  mixed  to  form  a  liquid  of  specific  gravity  1.25? 

8.  Two  boys  start  together  and  walk  around  a  circular  half-mile  track 
at  the  rates  of  3  V  and  4  miles  an  hour  respectively.  After  how  many  laps  will 
they  pass  each  other? 

9.  A  can  do  a  piece  of  work  in  3  days,  B  in  5  days.  How  long  will  it  take 
them  both  to  do  it? 

10.  A  can  do  a  piece  of  work  in  a  days,  B  in  6  days.  How  long  will  it  take 
them  both  to  do  it? 

11.  A  can  do  a  piece  of  work  in  a  days,  B  in  6  days,  and  C  in  c  days.  In 
how  many  days  can  they  together  do  it? 

12.  At  what  time  between  4  and  5  are  the  hands  of  a  clock  together? 

13.  At  what  time  between  10  and  11  are  the  hands  of  a  clock  at  right  angles? 
Opposite  each  other? 

14.  The  sum  of  the  ages  of  A,  B,  and  C  is  60  years.  In  how  many  years 
will  the  sum  be  5  times  as  great  as  it  was  10  years  ago? 

15.  Water  flows  into  a  cistern  through  two  pipes  A  and  B,  and  out  through 
a  third  pipe  C.  The  cistern  can  be  filled  by  A  in  1  hour,  by  B  in  45  minutes, 
and  emptied  by  C  in  36  minutes.  How  long  will  it  take  to  fill  the  empty  cis- 
tern when  all  three  pipes  are  running? 

61.  Simultaneous  Linear  Equations.  —  Let  there  be  given  two 
linear  equations  containing  two  unknown  quantities  x  and  y,  as 

ax  -\-  hy  -\-   c  =  0, 
a'x  +  h'y  +  c'  =  0. 

It  is  required  to  obtain  all  pairs  of  values  of  x  and  y  which  sat- 
isfy both  equations  simultaneously. 

First  Method  —  By  Substitution.  —  Solve  one  of  the  equations 
for  either  of  the  unknowns  in  terms  of  the  other;  substitute  the 
value  so  found  in  the  second  equation,  thus  obtaining  a  linear 
equation  in  one  unknown;  solve  for  this  unknown  and  then 
obtain  the  other  by  substitution  in  either  of  the  given  equations. 

Check.  Substitute  the  values  of  x  and  y  in  the  equation  not 
used  in  the  last  step  of  the  solution. 

Example.  Solve  for  x  and  y  : 

^^+x  =  15     and     ?^  +  y  =  6. 

Clearing  and  simplifying: 

4  X  -f  2/  =  45     and     x  -h  4  ?/  =  30. 


62] 


SIMULTANEOUS  LINEAR  EQUATIONS 


47 


From  the  first  of  these, 
Substituting  in  the  second 
Hence 
Then 

X  -  y 


Check. 


2/  =  45  -  4  x. 

z  +  4  (45  -  4  x)  =  30. 

15  X  =  150     or     X  =  10. 

?/  =  45  -  4  X  =  5. 

10 


+  y 


+  5  =  1+  5 


Second  Method  —  By  Elimination.  —  Multiply  the  first  equa- 
tion by  a',  the  second  by  —a,  and  add  the  resulting  equations 
together.  This  eliminates  x,  and  yields  a  linear  equation  in  y 
alone,  from  which  y  may  be  found.  Similarly  x  is  found  by  mul- 
tiplying the  first  equation  by  h' ,  the  second  by  -6,  and  adding. 
The  proper  multipliers  for  the  two  eliminations  are  conveniently 
indicated  thus: 


h' 

a' 

ax-hby  +  c 

=  0, 

-  h 

—  a 

a'x  +  h'y  +  c' 

=  0. 

Check.     Substitute  the  values  of  x  and  y  in  either  of  the  given 
equations. 

Example.     Solve  for  x  and  y  : 

8  X  -  15  y  +  30  =  0  and  2  X  +  3  !/  -  15 

Indicating  the  multipliers: 


3 

2 

8a:-15y  +  30  =  0 

15 

-8 

2x-\-    3  i/-  15  =  0 

0. 


<  -  54  ?/  +-  180  =  0,  or  y  =  V, 

5ix  -  135  =  0,  or  a:  =  i. 
Check.     8  X  §  -  15  X  V  +  30  =  20  -  50  -f-  30  =  0.  . 

62.  Exceptional  Cases. 

1.    The  given  equations  are  not  independent. 

In  this  case  one  equation  is  a  multiple  of  the  other,  so  that 
a  =  ka',  b  =  W,  and  c  =  k'c', 
k  being  a  constant.     Both  equations  are  then  equivalent  to  a  single 
one,  and  do  not  suffice  to  determine  two  unknowns. 

By  assuming  any  value  for  x,  substituting  in  one  of  the  equations 
and  solving  for  y,  we  obtain  a  pair  of  values  which  satisfy  both 
equations.  (Why?)  Hence  there  exists  an  infinite  number  of 
solutions. 


48  SIMULTANEOUS  LINEAR  EQUATIONS  [63,64 

2.    The  given  equations  are  inconsistent. 

If  a  =  ka'  and  h  =  W ,  but  c  j^  kc',  then  the  given  equations 
are  self-contradictory.  For  if  we  subtract  k  times  the  second 
equation  from  the  first,  we  obtain  c  =  kc',  which  is  not  true. 

In  this  case  there  is  no  finite  solution  possible.  For  if  we  assume 
X  =  xi  and  y  =  yi  to  he  a,  solution  of  either  equation,  the  other 
equation  will  not  be  satisfied  by  these  values  because  c  7^  kc'. 

63.   Graphic  Solution  of  Two  Simultaneous  Linear  Equations. 
Let  the  equations  be 

(1)  ax  -{-  by  +  c    =0, 

(2)  a'x  +  h'y  +  c'  =  0. 


The  graph  of  each  equation  is  a  straight  line. 
Suppose  Li  and  L2  (figure)  to  be  the  graphs  of 
equations  (1)  and  (2)  respectively.  Then  the  coordinates  of  any 
point  on  Li,  as  Pi,  satisfy  equation  (1),  and  of  any  point  P2  on 
L2  satisfy  (2).  Hence  the  coordinates  of  the  intersection  P  of- Li 
and  L2  satisfy  both  equations  simultaneously  and  give  the  required 
solution. 
Exceptional  Cases. 

1.  The  given  equations  are  not  independent. 

Then,  as  before,  a  =  ka',  b  =  kb',  and  c  =  kc'.  The  lines  Li 
and  L2  will  coincide  and  have  an  infinite  number  of  common  points. 

2.  The  given  equations  are  inconsistent. 

Then  a  =  ka',  b  =  kb',  but  c  7^  kc'.  The  lines  Li  and  L2  are 
now  parallel  to  each  other,  but  not  coincident.  Hence  they  have 
no  common  point  (except  at  infinity).  Including  the  infinite 
solution  is  equivalent  to  the  statement  "  parallel  lines  meet  at 
infinity." 

64.   Exercises.     Solve  for  x  and  y,  including   graphical   solu- 


tions: 


1.  2x  +  y  =  11. 
3x  —  2/  =  4. 

2.  3a;  +  8y  =  19. 
3x-y  =  1. 

3.  2X+7J  =47. 
X  +  y  =  15. 

4.  3x+42/  =  85. 
5a; +  4?/ =  107. 


5x  +  7y 

=  101. 

7  X 

-y  = 

55.1 

2x 

-y  - 

1  =0. 

Qx 

-32/ 

-3=0. 

15  X 

-Ty 

=  9. 

92/ 

-Ix 

=  13. 

2x 

-72/ 

=  8. 

42/ 

-9x 

=  19. 

65,66]  SIMULTANEOUS  LINEAR  EQUATIONS  49 


9. 

x-2y-^2  =0. 
3a; -62/ +  2  =  0. 

10. 

Bx  +  Zy  =  3. 
12a;  +  9?/  =  3. 

11. 

X  +  J  2/  -  3  =  0. 

12. 

3y-4x-l=0. 
18-3x  =  42/. 

13. 

2x  =  ll+92/. 
3x  -  15  =  V2y. 

14. 

2x4-72/ =  52. 
3x  -52/  =  16. 

15. 

3x-f42/-5  =0. 
^x  +  5  2/-i  =0. 

16. 

5  2/  -2x  =21. 

13x-42/  =  120. 

17. 

-+^  =  7 
2^3       ^• 

2x  +  3  2/  =  48. 

18. 

?^+^=34. 

¥+8^-^- 

19. 

4+.  =  ^^. 

.-s  =  V^. 

20. 

f  =  IO-|. 

4|2/  =  5x  -  7. 

Simultaneous  Linear  Equations  in  More  than  Two 
Unknowns 

65.  Three  Equations  in  Three  Unknowns. 

Let  the  given  equations  be, 

(1)  ax  +  hy  +  cz  +  d^  0, 

(2)  a'x  +  h'y  +  c'z  +  d'  =  0, 

(3)  a"x  +  h"y  +  c"z  +  d"  =  0. 

Eliminate  one  of  the  variables,  say  z,  from  two  pairs  of  the 
equations,  as  from  (1)  and  (2)  and  from  (2)  and  (3).  Solve  the 
resulting  equations  for  x  and  y.  Substitute  the  values  of  x  and  y 
so  found  in  one  of  the  given  equations  and  solve  the  result  for  z. 

Check.  Substitute  the  values  of  x,  y,  and  z  so  found  in  either  of 
the  equations  not  used  in  the  last  step  of  the  solution. 

66.  Exceptional  Cases. 

1.    The  given  equations  are  not  independent. 

(a)  In  this  case  one  of  the  equations  can  be  expressed  as  a  linear 
combination  of  the  other  two,  with  constant  coefficients.  Hence 
any  solution  of  these  two  equations  is  also  a  solution  of  the  third. 
But  two  equations  in  three  variables  admit  an  infinity  of  solutions. 
For  we  can  choose  any  value  for  z  at  pleasure,  substitute  it  in  the 
two  equations  and  obtain  a  pair  of  values  of  x  and  y. 


50  SIMULTANEOUS  LINEAR  EQUATIONS  [67-69 

(b)  It  may  happen  that  two  of  the  equations  can  be  expressed 
as  simple  multiples  of'the  third.  Then  any  solution  of  the  third 
equation  is  also  a  solution  of  the  other  two.  Hence  again  there 
exists  an  infinitij  of  solutions,  since  we  may  choose  for  two  of  the 
variables  any  value  at  pleasure  and  obtain  the  corresponding 
value  for  the  third. 

2.    The  equations  are  inconsistent. 

In  this  case  the  equations  in  x  and  y  obtained  by  eliminating 
z  are  also  inconsistent.  Hence  there  is  no  solution  (except  the 
infinite  .solution) . 

67.  We  shall  not  discuss  here  the  graphic  solution  of  three  linear 
equations  in  three  variables.  Interpreted  graphically,  each  of  the 
equations  (1),  (2)  and  (3)  represents  a  plane  in  space.  In  general, 
three  planes  meet  in  a  single  point,  giving  one  and  only  one  solu- 
tion.    The  exceptional  cases  are: 

1.  (a)  The  three  planes  meet  in  a  common  line.  Hence  any 
point  in  this  line  gives  a  solution. 

(b)  The  three  planes  coincide.     Hence  any  point  in  one  of 
the  planes  is  a  solution. 

2.  The  three  planes  are  parallel.  No  solution,  except  infinity. 
("  Parallel  planes  meet  at  infinity.") 

68.  Four  Equations  in  Four  Unknowns.  —  Solution.  Eliminate 
one  of  the  unknowns  from  three  different  pairs  of  the  four  given 
equations.  The  three  resulting  equations  can  be  solved  for  the 
other  three  unknowns.  The  fourth  unknown  is  then  found  by 
substituting  these  three  in  one  of  the  given  equations. 

Check.  Substitute  the  values  of  the  four  unknowns  in  one  of 
the  equations  not  used  in  the  last  step  of  the  solution. 

Exceptional  cases  arise,  quite  analogous  to  the  preceding.  We 
shall  not  discuss  them  here. 

The  method  of  solution  outlined  above  is  evidently  applicable 
to  any  number  of  linear  equations  in  the  same  number  of  variables. 
A  more  convenient  method  involves  the  use  of  determinants. 
(Chapter  XVI.) 

69.  Exercises  and  Problems. 

^'  4  +  6  "  ^^"-  ^-   3  +  2  "  3 

3      8       2"  2"'"3       a' 


69] 


EXERCISES 


51 


,    x+y      y -X 
^-  ~Y    +      2 


X      x  +  y 


=  5. 


4. 

£±I+L^t'.5. 

^-±y-^^.ro. 

5. 

V+'^^-- 

2x  +  =-V^^=21. 

6. 

i^^=.-.. 

2^-»+2»  =  i. 

16  2  (5  -  11  X)       11  -7?y 
"•  11  (x  -  1)    "^    3  -  2/ 
7  +  2x  _  125^  144?/  ^ 
3  —  X         30  (y  -|-  5) 

7-6x         4-3x 


17. 


18. 


19. 


107/ -19      5y-ll 
6x-10y-17  _^4x-Uy  -.5. 
3x-5y  +  2  ~2x-72/  +  12 

1 ^^  ^2 

l-x  +  ^      x  +  y-1      3' 

1 ^3 

1  4' 


1 

I -x+y       1 

1 


x  + 


7.  .25  X  +  3  ?/  =  10. 

4.5  X  -  4  7/  =  6. 

8.  4.2  !/  +  4  X  =  33. 
0.77  7/ -  0.3  X  =2.95. 

9.  0.2525  x  + 0.33  7/  =  280. 
3.122  x  + 0.055  7/  =  3096. 

10.  0.2 7/  +  0.25 x  =  2{y  -x) 
0.8  X  -  3.7  7/  =  -  15.3. 

11.  0.1 7/+ 0.3  X  =0.3. 
0.05?/  + 0.15  X  =0.15. 


12. 

|x-0.6  7/  = 
5  (x  -  1) 

0. 
5 

2  (2  7/ +  3) 

6 

13. 

x^y      30 

1       1  _   1 
X      7/      30* 

14. 

X           7/ 

15. 

1          '> 

~  +  -  =  10. 

x^y 

^(-^■- 

20. 

ax  —  by  =  m. 

cx+dy  =  n. 

21. 

x  +  y  =  3a-26. 

x-7/  =  2a-36. 

22. 

^^^■ 

X 

23. 

^"-f=c. 
a       b 

^^«■ 

24. 

^^''■ 

"+?- 

25. 

7HX  ,vy_i 

71   ^   q 

''-'M  =  v. 

n        s 

26. 

X  -  7/  +  1 

7/  -  X  +  1 

^ -^  =  7?in. 

X  -  y  +  1 


52  EXERCISES 


=  0. 
=  0. 

+  3=0. 
=  +1=0. 

0. 


35.   2  X  +  3  2/  =  12. 
3x  +  22  =  11. 

3  2/  +  4  z  =  10. 

li  X  +  n  y  =  10.  38.   x  +  2  ?/  +  3  z  =  32. 

25  X  +  2i  2  =  20.  2  X  +  3  y  +  2  =  42. 

3i  2/  +  3i  2  =  30.  3  X  +  2/  +  2  2  =  40. 

39.    U^=2.  40.   -^  =  1- 

y      z  X  +  2/      5 

'  +  '=*•  ^-s- 

X      z  X  +  z      6 

1    ,   1  ^g  _y^  ^1. 

X      2/         ■  2/^27 

41.    X  +  2  2/  =  5.  42.    2/  +  2  +  w  =  2.  43.  3  x  +  2/  +  z  =  4. 

2/  +  22=8.  2  +  M+x  =  3.  x+42/  +  3m  =  6. 

2  +  2u  =  ll.  m+x+2/  =  4.  6x+z  +  3u  =  8. 

M  +  2x  =  6.  x+2/+z  =  5. .  82/  +  32  +  5U  =  10. 

44.  Find  two  numbers  whose  sum  is  1735  and  difference  555. 

45.  If  at  a  given  place  the  longest  day  exceeds  the  shortest  night  by 
'8  hours  10  minutes,  what  is  the  duration  of  each? 

46.  The  sum  of  two  numbers  is  1000.  Twice  the  first  plus  three  times  the 
second  equals  2222.     Find  the  numbers. 

47.  The  annual  interest  on  a  capital  is  $180;  at  a  rate  of  interest  \\% 
higher,  the  annual  interest  would  be  $240;  find  the  capital  and  rate  of  interest. 

48.  A  farmer  sells  200  bushels  of  wheat  and  60  bushels  of  corn  for  $252; 
60  bushels  of  wheat  and  200  bushels  of  corn  would  bring,  at  the  same  price 
per  bushel,  $203;  find  the  price  per  bushel  of  each. 

49.  Two  points  move  on  the  perimeter  of  a  circle  999  ft.  long;  the  one  point, 
moving  four  times  as  fast  as  the  second,  overtakes  it  every  37  seconds;  find 
the  speed  of  each. 

50.  A  vat  of  capacity  450  cu.  ft.  can  be  filled  by  two  pipes.  If  the  first 
pipe  flows  3  minutes  and  the  second  1  minute,  40  cu.  ft.  are  discharged;  if  the 
first  pipe  flows  1  minute  and  the  second  7  minutes,  60  cu.  ft.  are  discharged. 


G9]  EXERCISES  53 

How  long  will  it  take  both  pipes  to  fill  the  tank,  and  what  is  the  discharge  per 
minute  of  each  pipe? 

51.  How  many  pounds  of  copper,  and  how  many  of  zinc,  are  contained  in 
124  pounds  of  brass  (alloy  of  copper  and  zinc),  if,  when  placed  in  water,  89 
lbs.  of  copper  lose  10  lbs.  in  weight,  7  lbs.  zinc  lose  1  lb.,  and  124  lbs.  brass 
lose  15  lbs.? 

52.  An  alloy  of  gold  and  silver  weighing  20  lbs.  loses  1|  lbs.  when  placed 
in  water.  How  much  gold  and  how  much  silver  does  it  contain,  if  gold,  when 
placed  in  water,  loses  I'g  of  its  weight,  and  silver  xV  of  its  weight? 

53.  Find  the  lengths  of  the  sides  of  a  triangle  if  the  sum  of  the  first  and 
second  is  30,  of  the  first  and  third  33  and  of  the  second  and  third  37. 

54.  Find  three  numbers  which  are  in  the  ratio  of  2  :  3  :  4  and  whose  sum 
is  999. 

55.  The  contents  of  three  measures  are  as  4  :  7  :  6;  10  measures  of  the 
first  kind,  4  of  the  second,  and  2  of  the  third  together  contain  20  gallons.  How 
much  does  each  measure  contain? 

56.  A  vessel  may  be  filled  by  each  of  three  measures  as  follows :  by  4  of  the 
first  and  4  of  the  third,  or  by  20  of  the  first  and  20  of  the  second,  or  by  28  of 
the  first  and  3  of  the  third.  Also,  the  three  measures  together  contain  29 
pints.     Find  the  content  of  each  measure. 

57.  A  vessel  can  be  filled  by  three  pipes:  by  the  first  and  second  in  72 
minutes,  by  the  second  and  third  in  2  hrs.,  and  by  the  first  and  third  in  1| 
hrs.     How  long  will  it  take  each  pipe  alone  to  fill  the  vessel? 

58.  A  and  B  can  do  a  piece  of  work  in  12  days,  B  and  C  in  20  days,  A  and 
C  in  15  days.  How  long  will  it  take  A,  B,  and  C,  working  together,  to  do  the 
job? 

59.  Three  principals  are  placed  at  interest  for  a  year,  A  at  4%,  B  at  5%, 
C  at  6%;  the  interest  on  A  and  B  is  $796,  on  B  and  C  $883,  and  on  A  and  C 
$819.     Find  the  amount  of  each  principal. 

60.  Two  bodies  move  on  the  circumference  of  a  circle;  when  going  in  the 
same  direction  they  meet  every  30  seconds,  and  when  going  in  opposite  direc- 
tions every  10  seconds;  in  the  second  case,  when  they  are  30  ft.  apart,  they 
will  again  be  30  feet  apart  after  3  seconds.  Find  the  speed  of  each  body  and 
the  radius  of  the  circle. 


CHAPTER  V 

Quadratic  Equations 

71.  Suppose  we  wish  to  find  two  numbers  whose  sum  is  5  and 
whose  product  is  6. 

I^et  X  =  one  of  the  numbers; 

then  3  —  a;  =  the  other  number, 

and  x{5  —  x)=  Q  or  x~  —  5x-{-Q  =  0. 

To  determine  x  we  must  solve  this  equation. 
Definition.     An  equation  of  the  form 

rtic^  -\-bx  -\-  c  =  0, 

where  a:  is  a  variable  and  a,  b,  c  are  constants,  is  called  the  gen- 
eral equation  of  the  second  degree  in  07ie  variable,  or,  a  quadratic 
equation  in  x. 

Methods  for  Solving  the  Equation  ax^  +  6x  +  c  =  0. 

72.  1.  By  Factoring.  When  the  trinomial  ax^  -\- bx  -\-  c  can 
readil}^  be  factored,  then  each  factor,  equated  to  zero,  gives  a 
value  of  X. 

Example.  x2  -  5  x  +  6  =  0,    . 

or  (X  -  2)  (x  -  3)  =  0. 

X  -  2  =  0     or     X  -  3  =  0. 
Hence  x  =  2     or     x  =  3. 

73.  2.   By  Completing  the  Square. 

(a)    The  equation  is  reduced  to  the  form 
(x  +  /i)2  =  k, 
whence  x  +  h  =± Vk,  and   x  =-  h  ±Vk. 

This  reduction  is  effected  as  follows  : 

Given  ax-  +  fex  +  c  =  0. 

Transpose  c  :  ax-  +  bx  =  —  c. 

54 


73]  •      QUADRATIC  EQUATIONS  65 

.   ,   b  c 

Divide  by  a:  ^*' "^  a^  ^  "  a' 

Add  l^j  to  both  members: 

h       ,   /  b\-  c    ,   /  h 


or, 


a  \2a)  a       \2a 


^Ll 

62- 

4 

4ac 
a2 

W^ 

—  4ac 
4a2 

•=± 

-b±' 

v/6«- 

4ac 

-+2^-±V^T^=±2^^'''^'-*- 


a 


Hence, 

(b)    The  equation  is  reduced  to  the  form  (2  ax  +  h)-  =  k, 


whence           2  ax 

-f  /i  =  -t  Vk,   and  x  = ^ 

z  a, 

Given 

ax"  +  hx-}-c  =  0. 

Transpose  c: 

ax^  +  bx=-  c. 

Multiply  by  4  a: 

4  a2a;2  +  4  a6a;  =  -  4  ac. 

Add  62:                4 

,  ^2^2  _^  4  (j^l)j^  _|_  ^2   =  52  _  4  ^f.^ 

or, 

(2  ax  +  6)2  =  62-4  ac. 

2  ax  +  6  =  ±  ^62  -  4  ac. 

-  6  ±  V62  -  4  ac 

Hence, 

2a 

Example. 

2  2;2  +  X  -  G  =  0. 

(a)  Transpose -6: 

2  x2  +  X  =  6. 

Divide  by  2: 

x2  +  i  X  =  3. 

Add  (1)2: 

x2  +  ^x  +  a)2  =  3+a)2, 

or 

(a;  +  \y  =  n- 

x+\ =±h 

and 

X  =±l  -I  =  I   or    -2. 

(b)  Transpose  -6: 

2  x2  +  X  =  G. 

Multiply  by  8: 

IG  x2  +  8  X  =  48. 

Add  12or  1: 

IG  x2  +  8  X  +  1  =  49, 

or, 

(4  X  +  1)2  =  49. 

4  X  +  1  =  ±  7. 

Hence 

X  =  i   or  —  2,  as  before. 

56  QUADRATIC   EQUATIONS  [74-76 

74.   3.   By  Formula.     In  (72),  by  completing  the  square  accord- 
ing to  either  method,  we  obtained 


—  b  ±  Vb^  —  ^ac 

Of.  =  _ 

2  a 

Any  quadratic  equation  in  x  may  be  solved  directly  by  means 
of  this  formula,  by  merely  inserting  for  a,  h,  and  c  their  values 
from  the  given  equation.  The  formula  should  be  carefully  com- 
mitted to  memory. 

Example.   2  x2  +  x  -  6  =  0. 

-  l±Vl-4X2X( 


(-< 

^  = 

-  1  ±  7      3 
4             2 

or   — 

2. 

11. 

5x 

(X  -  2)  +  i  = 

1  -3 

X. 

12. 

(1 

-3x)  (x-6)  = 

=  2(x 

+  2). 

13. 

{X 

+  l)(2x  +  3) 

=  4x2 

-22. 

14. 

Ix 

2  -  48  =  2  X  (x 

+  7). 

15. 

13 

x2  -  30  =  6  (1 

-X)2 

+  63. 

16. 

ax 

+  b  =  X2. 

17. 

bx 

-  2  62  +  x2  =  ; 

2  6x. 

18. 

X' 

+  mn  =  —  (m  • 

■fn)x 

19. 

ex 

-2c-x2  =- 

-2x. 

20. 

X2 

ax      a2 

2   ~  2' 

4 

75.  Exercises.     Solve  for  x: 

1.  x2  +  4  X  -  12  =  0. 

2.  x2  -8x=  -7. 

3.  x2+6x  =  16. 

4.  x2  +  12  =7x. 

5.  14  =  x2  -5x. 

6.  5x2  _3x-2  =0. 

7.  3x2+5x  -42  =0. 

8.  3x2  _50  =25x. 

9.  2x2 -13x  =-15. 
10.  3x2  _7x  -6  =  0. 

76.  Definition.  A  root  of  an  equation  is  a  value  of  the  variable 
which  satisfies  the  equation. 

By  the  formula  of  (73),  the  two  roots  of  any  quadratic  equa- 
tion can  be  obtained. 

Nature  of  the  roots  of  the  equation  ajc"^  +  6ac  +  c  =  0.  —  The 
values  of  x  obtained  by  the  formula 


X  = 


h  ±  Vb^  -4ac 
2a 

will  be 

1.  real  and  unequal  if  b^  ~  4  ac  >  0; 

2.  real  and  equal      if  b^  —  4  ac  =  0] 

3.  imaginary  if  &^  -  4  ac  <  0. 


77-78]  QUADRATIC  EQUATIONS  57 

For,  in  the  first  case  the  radicand  in  the  formula  for  x  is  positive, 
hence  the  square  roots  are  real;  in  the  second  case,  the  radicand 
vanishes,  and  the  two  values  of  x  reduce  to  the  common  value 
—  6  -i-  2  a;  in  the  third  case  the  radicand  is  negative,  hence  both 
square  roots  are  imaginary. 

The  expression  6^  —  4  ac,  on  whose  value  depends  the  nature  of 
the  roots,  is  called  the  discrhninant  of  the  equation  ax-  -{-bx-{-c  =  0. 

When  the  discriminant  vanishes,  the  roots  are  equal;  ax^  +  6x  +  c 
is  then  a  perfect  square. 

77.  Exercises.  —  Without  solving  the  equations,  determine  the 
nature  of  the  roots  of: 

1.  Exercises  1-10  of  (74).  6.  ^x^-hx-i^O. 

2.  4x2+4x  +  l=0.  7.  0.1x2+0.5x+0.8  =0. 

3.  x^+x  +  l  =0.  8.  1^  x2  -  6J  x  +  8}  =  0. 

4.  6x2 +2X-1  =0.  .         9.  ^^2 -ix  +  ?j  =0. 

5.  9  x2  +  12  X  +  4  =  0.  10.  0.06  x2  +  0.22  x  +  0.08  =  0. 

For  what  values  of  the  Hteral  quantity  involved  in  the  following  equations 
will  the  roots  be  real  and  unequal,  equal,  or  imaginary  respectively : 

11.  x2 +2x+c  =  0.  18.  2x2+4/ix -/i2  =0. 

12.  4 x2  +  4 X  +  ;i  =  0.  19.  2x2  +  4 ax  -  a  =  0. 

13.  3x2  -  2x  -  A;  =  0.  •    20.  ax2  +  2x  +  1  =  a. 

14.  ^  x2  -  J  X  +  4  a  =  0.  21.   a2x2  +  ax  +  5  =  0. 

15.  x2  +  2  ?jx  +  4  =  0.  22.   2  cx2  +  3  X  -  c2  =  0. 

16.  3x2  -  4/cx  +  5  =  0.  23.    (l  +  k)  x^  +  x  -  k  =  0. 

17.  6  x2  +  -  X  -  3  =  0.  24.  -  +  ^  X  +  — ^  =  0. 

7n  n       2         n  +  1 

78.  Relations  between  the  Coefficients  and  Roots  of  a  Quadratic 
Equation.  —  The  roots  of  the  equation 

ax^  -f-  6x  +  c  =  0 


Xi  = 


Hence 


_  5  _^  V62  -  4  ac             -  6  -  V6-'  - 

-4ac 

2  a               '  ^-                   2  a 

Xi  +  X2  = '     and     X1X2  =  -• 

a                             a 

58  QUADRATIC  EQUATIONS  [79-81 

That  is,  if  the  equation  he  divided  by  the  coefficient  of  x^,  the  new 
coefficient  of  x,  with  its  sign  changed,  equals  the  sum  of  the  roots; 
the  new  constant  term  equals  the  product  of  the  roots. 

79.  Factors  of  the  Trmomial  wx^  +  &x  +  c.  —  If  xi  and  X2  be 
the  roots  of  the  equation  ax-  +  bx  +  c  =  0,  the  trinomial  is  divis- 
ible by  X  —  xi  and  x  —  X2.     But 

(x  —  xi)  {x  —  X2)  =  X-  —  {xi  -\-  X2)  x  +  a;ia;2 

.  ,  6      ,  c 

=  x~  -\-  -  X  -\ 

a         a 

Therefore  a{x  —  xi){x  —  Xo)  =  ax^  +  bx  +  c. 

Hence  to  factor  the  trinomial  ax-  -\-  bx  -\-  c,  place  it  equal  to 
zero  and  solve  for  x;  subtract  each  root  from  x,  form  the  prod- 
uct of  these  differences  and  multiply  it  by  a. 

80.  Exercises. 

1.  Find  the  sum  and  the  product  of  the  roots  of  the  equations  in  exercises 
1-10  of  (76). 

Form  equations  whose  roots  are: 

2.  2,  3;  4,  -1;   -2,  -1. 

'    3.    a,  2  a;  p,  q;  m  +  n,  m  —  n. 

4.    V^^,  -  V^T;  1  +  V^^,  1  -  V^^;  a  +  b  V^,  a-b  \f^^. 
5-14.    Factor  the  left  members  of  the  equations  in  exercises  1-10  of  (74). 
15-24.    Same  for  exercises  1-10  of  (76). 

25.  Show  that  the  equation  y  =  x-  +  bx  +  c  cannot  have  a  fractional  root 
if  b  and  c  are  integers. 

81.  Graphic  Solution  of  Quadratic  Equations.  —  In  order  to 
solve  the  equation 

ax^  +  bx  -\-  c  =  Q, 

we  must  find  the  values  of  x  which  reduce  the  trinomial  ax^  + 
bx  -\-  c  to  zero.  When  a,  b,  c  are  given  numerical  values,  the 
required  values  of  x,  when  real,  may  be  obtained,  exactly  or 
approximately,  by  trial. 

Consider,  for  example,  the  equation 

2  x2  +  a;  -  6  =  0. 


S2,S3] 


QUADRATIC  EQUATIONS 


59 


0, 

+  1, 

+  2, 

+  3, 

G, 

-3, 

+  4, 

+  15 

Designate  the  trinomial  on  the  left  by  y,  so  that 

y  =  2x^  +  x-6. 

Now  form  a  table  showing  the  values  of  y  corresponding  to  a 
series  of  assumed  values  of  x : 

x=   .  .   .   -3,    -2,    -1, 

y=    .   .   .   +9,        0,-5, 

We  see  that  y  =  0  when  x  =  —  2,  which 
gives  one  root  exactly.  Also,  y  must  be  zero 
again  for  a  value  of  x  between  +1  and  +2, 
hence  the  other  root  lies  between  1  and  2. 

Now  consider  the  pairs  of  corresponding 
values  of  x  and  y  as  the  coordinates  of  a 
series  of  points  and  draw  a  smooth  curve 
through  them  (figure).  Scaling  oflf  the  values 
of  X  for  which  y  =  0,  we  have 

x  =  —  2  and  a:  =  1.5  approximately. 

82.  Parabola.  —  The  curve  in  the  figure  is 
called  a  parabola.  It  is  an  example  of  a  class 
of  curves  all  of  which  have  similar  forms.  The  point  where  the 
curve  bends  most  sharply  is  its  vertex,  and  a  line  through  the 
vertex  and  dividing  the  cUrve  into  two  sym- 
metrical portions  is  called  the  axis  (figure). 
The  segments  OA  and  OB,  measured  from  the 
origin  to  the  points  where  the  curve  cuts  the 
X-axis,  are  called  the  ic-intercepts.  The  inter- 
cepts are  positive  when  extending  to  the  right 
from  0,  negative  when  extending  to  the  left. 
-^83.  It  will  be  found  that  the  graph  of  the 
equation 


r 

o 

X 

1 

V 

/ 

zj  =  2x'  +  X 


\ 

Y 

5 

/ 

A 

O 

A    X 

\ 

\ 

•^ 

I 

Parabola 


y  =  ax.^  -\-  bx  -\-  c 
with  its  axis  parallel  to  the  ?/-axis.     (We 


is  always  a  parabola 
assume  a  ?^  0.)    » 

The  parabola  will  cut  the  x-axis  in  two  distinct  points,  or  be 
tangent  to  the  a:-axis,  or  will  not  cut  the  a:-axis  at  all  according 
as  the  equation  ax'^  -\-  hx  -{-  c  =  Q  has  real  and  unequal,  or  equal, 
or  imaginary  roots.     For  in  the  first  case  y  is  zero  for  two  distinct 


60 


QUADRATIC  EQUATIONS 


[84,85 


values  of  x,  in  the  second  for  two  equal  values  of  rr,  and  in  the 
third  for  no  real  value  of  x. 

These  three  cases  are  illustrated  in  the  figures  below. 


IS 


Y 

r) 

\ 

1 

\ 

\ 

1 

\ 

0 

\ 

y 

X 

._! 

9 

-2 S 


=  x*  —  4x  +  3      y  =  x'  —  4x  +  4 
V-AaOO  6^-4ac  =  0 


y  =  x^  — 4x  +  5 
V-  4ac<0 


84.  Exercises. 

1-10.    Solve  graphically  the  equations  in  exercises  1-10  of  (74). 
11-19.    Draw  the  graphs  representing  the  left  members  of  the  equations  of 
exercises  2-10  of  (76). 

20.  On  the  same  diagram  construct  the  graphs  of?/  =  x2+2x,  y  =  x'^-\-2x-\-\, 
and  2/  =  x2  +  2  x  +  2. 

21.  Same  as  in  20  for  ?/ =  -x2-2x,  y=  -x2-2x-l,  and  ?/=-x2-2x-2. 

22.  What  is  the  effect  on  the  graph  of  y  =  ax2  +  6x  +  c  when  c  is  increased 
or  diminished? 

23.  What  is  the  efifect  on  the  graph  of  changing  the  signs  of  all  terms  of  the 
trinomial? 

85.  Equations  Reducible  to  Quadratics. 

Example  1.     2  x^  -  7  x2  +  6  =  0. 
Solve  for  x^  as  the  unknown  quantity. 


^,_7±V49-48^2or    | 

X  =  ±  V2    or    ±  i/|- 

Example  2. 

x-^  -8x-*  =9. 

Solve  for  x" 

"  *  as  the  unknown. 

,-J.8±>0,9„,        I. 

1 

X  =  ;,}^    or    -  1. 

729 


86]  QUADRATIC  EQUATIONS  61 

Example  3.     (2  x2  +  5  x)2  -  6  =  2  x2  +  5  x. 
Solve  for  2  x2  +  5  X  as  the  unknown. 

(2  x2  +  5  x)2  -  (2  x2  +  5  x)  -  6  =  0. 

2  x2  +  5  X  =  ^-^  =  3  or    -  2. 

2x2  +  5x=3  or  2x2  +  5x=  -2. 

X  =  1  or  —  3;  or,   X  =  —  I   or    —  2. 

Exercise.     Verify  the  answers  in  the  above  examples  by  substitution. 


Example  4. 

X  +  V2  x2  +  1  =  1. 

Transpose: 

V2  x2  +  1    =  1   -  J 

Square  and  collect  terms: 

x2  +  2  X  =  0. 

Therefore 

X  =  0  or 

Example  5. 

X-  V2x2  +  1  =  1. 

Transpose: 

-  V2  x2  +  1  =  1  -  ; 

Square,  etc.: 

x2  +  2  X  =  0. 

Therefore 

X  =  0  or 

Exercise.     Ve 

rify  the 

answers  in  examples  4  ar 

2,  as  in  example  4. 


On  substituting  the  values  found  in  examples  4  and  5  in  the  given 
equations,  we  find  that  the  first  equation  is  satisfied  by  both 
values  of  x,  but  not  the  second,  provided  we  assume,  as  usual,  that 
V2  x^  +  1  stands  for  the  positive  square  root. 

The  equation  of  example  5  may  be  put  in  the  form 


X  -  1  =  '\/2x-  -\-l. 

Evidently  these  two  expressions  are  not  equal  to  each  other  for 
any  real  value  of  x.  For,  if  x  be  less  than  1,  they  are  of  unlike 
signs;  if  x  be  greater  than  1,  V2  a;^  is  certainly  greater  than  x, 
and  therefore  V2  x^  +  1  >  a:  —  1.  Hence  the  solution  of  ex- 
ample 5  as  above  has  led  to  incorrect  results. 

The  reason  for  this  is  that  on  squaring  in  the  second  step  of  the 
solution  the  sign  of  the  radical  disappears,  and  from  that  point 
on  we  are  really  solving  example  4  also. 

When  an  equation  is  squared  to  clear  of  radicals,  the  answers  should 
be  carefully  verified  and  only  those  retained  which  satisfy  the  given 
condition. 

).  Exercises  and  Problems. 

"5  =  VSx  +  l.  3.   V2x2  _5x  +  l  =  ^x  +  l. 

V3(5x  +  7).  4.   4 X  -  1  =  V7x2  -2x  +  4. 


62  QUADRATIC  EQUATIONS.     EXERCISES 

6.  ^Jx  +  3  +  Va;+8  =  5  V^ 


6.  V2x  +  l  +  V7x-27  =  V3a;+4. 

7.  Vx+^  +  V3x  -3  =  10. 


8.  Va;  +  17  +  Va;  -4  =  I  V2x. 

9.  V2a;  +  1  +  Va:  -  3  =  2  V^- 
10.   V12+X  =  V7a;+8  -  2. 

4  4 


15 


16. 


17. 


2/  + V4 
5 


y  -  ^/4  -y^ 

5 

X  +  \/x^  +  5      X  -  Va;2  +  5 

4^=^  =  1+^(757-1). 
VS  2  +  1  2 


11. 
12. 
13. 
14. 

12_ 

'  7 


Vx2  +  X  +  3     ^3_ 
V2x2+5x-3  ~4 
V3x2+x  +  5  ^3_ 
V4x2  -x  +  1      2 
V9x2+6x  +  l    ^3_ 
Vl8x2-3x-2      2 

V9 x2  +  6 X  +  1    ^  _  3. 
Vl8x2-3x-2  2 


18.     ^/       ^    +  3  V2  y  +  1  =  7  V«^. 


19. 


V2t^  +  1 


V3s  + 


V5s  =  V3s  +  1. 


20.   V7iT4  +iUiii  =  7  V4^ 


21. 


V4< 
V3  x2  +  1  -  V2  x2  +  1  _1 


V3  x2  +  1  +  V2  x2  +  1       7 
(Or,  by  composition  and  division  rationalize  the  denominator.) 

22    V27  x2  +  4  +  V9  x2  +  5  ^^ 
'  V27  x2  +  4  -  V9  x2  +  5 

„«    %/5x 


■4  + 


X  _  Vix  +  1 
VSx  -  4  -  VS  -  X       V4 X  -  1 


24. 


V2 


V2 


X  Vx  +  X2  1  + 


\/x2  -  16 


+  Vx  +  3  = 


14. 

7 


86] 


QUADRATIC  EQUATIONS.     EXERCISES 


63 


30. 


X  +  m  _p  —  X 
X  —  m      p  +  X 

n  —  X  _  X  +  p 
n  -\-  X      X  —  p 

a^       X  X 


X  ft2         a2  _  ft2 

ab  —  X  _  b  —  ex 
b  —  ax      be  —  X 


36. 


31.  ?i±i^+^iziA^  =  2x4. 


1  ^• 

l2  -  X2         X        a 
X              b       X 

-  7n*       y^  —  m^ 

10  n 
Va. 

2/  ■ 
I  y/a 

-m         y  +  m 
.  +  X  -  Va  -  X  = 

.    m  - 

-  V2  my  -  y2 

m  +  V2  /ny  -  r/2 

;.  Vx 

+  V2a-x  =~ 
\Jx 

37, 


y/x  +  '\/b      Va  -  X  +  VT 


Vj  —  Vfe       \la  —  x  —  \lb  —  : 
.   a/t  +  Va  -  V^  +  a;2  =  Va- 


X  +  V  —  («-  +  ax) 


\Ja  — 


40. 


+  Va 


Vo  —  X  —  V^  —  b 

/I 


la  —  X 


41.  2  X  Va:  -  3 

^«      a  -  X 

42.  — F= 


X  -6 


=  20. 


V:t 


43.  l/^  +  iA-+^  =  . 
\'  6  +  X       y  o  —  X 

44.  v/F^-v/^^  = 
yb-x      ya  —  X 

45.  x'^'  -16x^  =512. 


46.  x2">  +  2  ax'"  =  8  a2. 

47.  x2  4-  V5x  +  x2  =  42  -  5 z. 

48.  Vx -5Vx2=  -  18. 

49.  7  V^^  +  V.^2  =  _  12. 


50.  x2+24=  7x-  Vx2-7x+18. 
51.  ^(TT^^-  ^V(i^^  =  ^/^^^ 

62.    Find  three  consecutive  integers,  the  sum  of  whose  squares  is  1202. 

53.  Find  three  consecutive  even  integers,  the  sum  of  whose  squares  is  776. 

54.  The  sum  of  the  squares  of  three  consecutive  integral  multiples  of  4  is 
3104.     Find  the  numbers. 

55.  A  rectangle,  twice  as  long  as  it  is  wide,  has  an  area  of   1800  square 
feet.     Find  its  dimensions. 

56.  How  large  a  square  must  be  cut  from  each  corner  of  a  rectangular 
card  6  X  12  inches  so  that  the  remaining  piece  shall  contain  27  square  inches  ? 

57.  As  in  56,  except  that  the  original  dimensions  are  aXb  inches  and  the 
remaining  area  A  square  inches. 

58.  What  changes  must  be  made  in  the  dimensions  of  a  rectangle  2  X  12 
inches  to  double  the  area  without  changing  the  perimeter  ? 


64  SIMULTANEOUS  QUADRATICS  [87,88 

59.  As  in  58,  when  the  original  dimensions  are  a  X  b  inches. 

60.  State  some  values  of  a  and  b  for  which  exercise  59  is  impossible. 

61.  Find  the  radius  of  a  cylinder  whose  height  is  10  feet,  if  the  total  sur- 
face in  square  feet  must  equal  the  volume  in  cubic  feet. 

62.  As  in  61,  except  that  total  surface  equals  twice  the  volume. 

63.  As  in  61,  except  that  total  surface  equals  n  times  the  volume.  For 
what  values  of  n  is  the  problem  impossible  ? 

64.  What  number  exceeds  twice  its  square  root  by  3  ? 

65.  The  sum  of  the  ages  of  a  father  and  his  son  is  80  years  and  the  product 
of  their  ages  is  15  times  the  sum;  find  the  age  of  each. 

66.  A  number  consisting  of  two  equal  digits  is  3  less  than  4  times  the 
square  of  one  of  its  digits;  find  the  number. 

67.  For  what  real  values  of  x  is  x^  +  10  x  +  9  positive  ?  zero  ?  negative  ? 
(Graph.) 

68.  Show  that  6  +  2  a  +  a^  cannot  be  negative  if  a  is  real.     (Graph.) 

69.  Show  that  3  a  —  a^  —  5  cannot  be  positive  if  a  is  real.      (Graph.) 

70.  The  difference  of  the  cubes  of  two  consecutive  integers  is  127.  What 
are  the  integers  ? 

71.  Two  trains  start  from  a  station,  one  going  due  north  5  miles  an  hour 
faster  than  the  other,  which  goes  west;  at  the  end  of  four  hours  they  are  60 
miles  apart.     Find  the  speed  of  each. 

87.  Simultaneous  Quadratics. 

Definition.  The  degree  of  a  monomial  involving  one  or  more 
literal  quantities  is  the  sum  of  the  exponents  of  such  literal  quan- 
tities as  may  be  specified. 

For  example  a%™y"  is  of  degree  m\n.x,nmy,m  +  n  in  x  and  y, 
m  -\-  n  -\-  y  m  a,  X  and  y. 

The  degree  of  a  polynomial  is  that  of  its  term  of  highest  degree. 

A  quadratic  equation  in  several  variables  is  one  in  which  all  the 
variable  terms  are  of  the  first  or  second  degree,  at  least  one  term 
of  the  second  degree  being  actually  present. 

88.  Solution  of  Two  Simultaneous  Equations  in  Two  Variables, 
one  being  Linear,  the  other  Quadratic.  —  The  most  general  forms 
of  such  equations  are: 

(1)  px  +  qy  -\-  r  =  0, 

(2)  aa;2  +  by^  -\-  cxy  -\-  dx  -\-  ey  -{-  f  =  0. 

Solution. 

1.  Solve  (1)  for  one  of  the  variables  in  terms  of  the  other.     Thus : 

px  +  r 
y  =  —  ' 

2.  Substitute  this  value  in  (2),  obtaining  a  quadratic  equation 


89-91]  SIMULTANEOUS  QUADRATICS  65 

3.  Solve  this  quadratic  for  x,  and  let  its  roots  be  Xy  and  X2. 

4.  The  corresponding  values  of  y  are  now  found  by  substituting 
these  values  for  x  in  the  first  step.     Thus: 

Tpxi  +  r           ,                   vx2  +  r 
Vi  =  —  ^-— — — '     and     7/9  =  —  ' ■• 

Example.  (a)      x  +  y  =  1, 

(b)     a:^  +  2/2  =  4. 
From  (a),  y  =  I  —  x. 

Substituting  in  (b) :     x'- +  {1  -  xy  =  4     or     2  x^  -  2  x  -  3  =  0. 
Hence  xi  =  5  +  i  V7;  xa  =  5  —  i  V"- 

Then  ^1  =  ^  -  i  V7;  2/2  =  I  +  W7. 

Reducing  to  decimals,  we  have  approximately 

(xi,  2/1)  =  (  +  1.8,  -0.8)     and     (x2,  2/2)  =  (-0.8,  +1.8). 
In  this  case  there  are  two  distinct  real  solutions. 

89.  Nature  of  the  Solutions  of  Equations  (1)  and  (2)  of  (88).— 

The  values  xi  and  X2  obtained  in  the  third  step  of  the  solution  in 
(88)  are  either  real  and  unequal,  real  and  equal,  or  both  imaginary. 
Then  the  values  of  y  obtained  in  the  fourth  step  will  be  of  the  same 
nature  as  the  values  of  x. 

Hence  there  are  always  two  solutions,  which  may  he  real  and  un- 
equal, real  and  equal,  or  imaginary. 

90.  These   three  cases  may  be  illustrated   by  means  of  the 
equations, 

(1) 

(2) 
Then       x^  +  {k  -  xY 
Hence     xi  =  h  {k  +  V 

yi  =  i{k-  VS-F)     and     2/2  =  U^'  +  Vs"^^/^). 
These  solutions  will  be 

real  and  unequal  if  A-  <  8; 
real  and  equal  if  A;- =8; 
imaginary  if  k^  >  8. 

91.  Graphic  Solution  of  the  equations 

(1)  x-hy=l, 

(2)  x^  +  y^  =  4. 


x  +  y=k, 

x'  +  t  =  4. 

=  4,     or     2x?  ■ 

-2kx-{-  (k- 
X2  =  h(k- 

--4) 
V8- 

=  0. 

'8-k'')     and 

k-n. 

66 


SIMULTANEOUS   QUADRATICS 


[92 


r 

/ 

K 

~~^ 

N, 

/ 

X 

X 

\ 

V 

\ 

y 

' 

Straight  line   x  +  y 
Circle  x^  +  if' 


Considering  x  and  y  as  the  coordinates 
of  a  variable  point,  all  values  of  x  and  y 
which  satisfy  the  equation  (1)  give  rise  to 
a  series  of  points  lying  on  a  straight  line 
(figure). 

Let  us  now  mark  some  points  whose 
coordinates  satisfy  equation  (2),  which  we 
put  into  the  form 


2/  =  ±  V4  -  X". 

Assuming  a  set  of  values  for  x,  and  calculating  the  corresponding 
values  of  y,  we  have 

a:  =  0,  i,  1,  U,  2,  2^,  .  .  .   ; 
y  =  ±2,±h  Vi5,  ±  V3,  ±  ?;  V7,  0,  imaginary. 

For  negative  values  of  x  we  obtain  the  same  values  of  y  over  again. 

On  plotting  these  values  we  obtain  a  series  of  points  all  of  which 
lie  on  a  circle  of  radius  2,  center  at  the  origin. 

The  points  of  intersection  of  the  line  and  the  circle  have  coordi- 
nates which  satisfy  both  equations  at  once,  and  are  therefore  the 
required  solutions.     Scaling  them  off  from  the  figure  we  have 

(:ri,  2/0  =  (1.8, -0.8)     and     (0:2, 2/2)  =  (-1-8, +0.8),  as  in  (88). 

92.  Graphic  Illustration  of  the  Three  Cases  of  (90).  —  In  (1) 
of  (90),  let  us  put  successively  A;  =  1,  2  V2,  and  4,  so  that  li-  <  8, 
=  8,  and  >  8  respectively.     We  have  then  the  equations, 

(1)  a;  +  y  =  1;   a:  +  2/  =  2  V2;   X  +  2/  =  4, 

(2)  a;2  +  i/^  =  4;   x2  +  2/2  =4;      x^  +  if  =  A. 

The  three  straight  lines  and  the 
circle  are  shown  in  the  adjacent  figure. 
When  k  =  1,  the  line  cuts  the  circle  in 
two  distinct  points;  when  k  =  2  V2, 
the  line  is  tangent  to  the  circle;  when 
A;  =  4,  the  line  fails  to  meet  the  circle. 
We  may  consider  these  three  cases 
as  arising  from  special  positions  of  a 
variable  line  which  moves  parallel  to 
itself  and  occupies  in  turn  the  posi- 
tions of  the  three  lines  in  the  figure.  Circle  x-+  y- 


r\ 

\ 

\ 

\ 

y 

\ 

--^ 

\ 

\ 

1 

0   \ 

\ 

\ 

\ 

\1 

\ 

y 

93,94] 


SIMULTANEOUS   QUADRATICS 


67 


93.   Standard  Equation  of  the  Circle. — 

The  equation 

x-  +  if  =  r' 

is  satisfied  by  the  coordinates  of  every 
point  on  a  circle  of  radius  r,  center  at  the 
origin,  and  by  no  other  point.     It  is  called 

the  standard  equation  of  the  circle. 

x'  +  y-  =  7^ 
Exercises.     Solve  for  x  and  y,  and  check  care-      Circle,  radius  r,  center  at 
fully  by  graphs. 


,x2  +  2/2 


(x-y  =0. 
^'  U-y  =  2. 


4. 


(X2+2/2 

\2x  +  y 


7. 


(  X2  +  J/2   =  4, 


origin 
X2  +  2/2  =9, 

\3x  +  4y  =  12. 

[x2+2/2  =9, 
!  4  X  -  5  2/  =  20. 

;4a;2  +  42/2  =  1, 
|3x-2/  =  l. 


(x2+2/2  =  1,  (x2  +  2/2  =  16, 

*  (x-2/  =  ^A2.  (2x  -3?/  =  4. 

10.  Determine  A;  so  that  the  line  x  +  y  =  k  shall  be  tangent  to  the  circle 
a;2  +  2/2  =  4. 

11.  Determine  m  so   that  the  line  y  =  mx  +  5  shall   touch   the    circle 
x2  +  2/-  =  5. 

12.  As  in  11,  for  the  line  y  =  mx  +  2  and  the  circle  x-  -\-  y-  =  f. 

94.    Consider  the  equations 

x-y  =  1, 

9^4 


Proceeding  as  in  (88),  we  obtain 


Xi 


2/1 


+  12  V3^ 

13 
-4  +  I2V3 
13 


2.3 


12%/^ 


1.3 


13 
-4 


0.9 


12  V3 


13 


1.9 


Graphic  Solution.  —  All  values  of  x  and  y  which  satisfy  the 
first  equation  are  the  coordinates  of  points  on  a  straight  line. 
We  now  plot  a  series  of  points  whose  coordinates  satisfy  the 
second  equation,  which  we  solve  for  y  in  terms  of  x  and  write  in 
the  form 

y=±\  V36-4z^'. 

Whena:=-3,        -2,         -1,       0,         +1,         +  2,   +  3, 
y=      0,  ±  3  Vo,  ±  5  V'2,  ±  2,  ±  ^  V2,  ±  §  Va,        0. 


68 


SIMULTANEOUS   QUADRATICS 


[95,96 


/ 

B  J 

/ 

^ 

"^ 

X^' 

( 

/ 

X 

A 

\ 

" 

/ 

A 

^s 

^^ 

/ 

^ 

/ 

X,Y., 

k 

32+2^ 


On  plotting  these  points  and  draw- 
ing a  smooth  curve  through  them  we 
obtain  the  curve  in  the  adjacent  figure, 
called  an  ellipse.  The  line  A' A  is 
called  the  major  axis  of  the  ellipse,  B'B 
the  minor  axis,  and  0  is  the  center. 
In  this  case,  A' A  =  Q  and  B'B  —  4; 
OA  =  3  and  OB  =  2. 

Scaling  off  the  coordinates  of  the 
points    of    intersection    of    the    two 


Ellipse 

Straight  line    x  —  y  =  1 
graphs,  we  have  as  our  graphic  solution 

(xi,yi)  =  (2.3,  1.3);    (x2,y^^  =  (-0.9, 

95.  Standard  Equation  of  the 
Ellipse.  —  Every  equation  of  the 
form 

represents  an  ellipse,  whose  major 
axis  is  2  a,  minor  axis  2  h,  center  at 
the  origin.  It  is  calle'd  the  standard 
equation  of  the  ellipse. 

Exercises.     Solve  and  check  by  graphs  : 


1, 


.1 


rx2     ,      2/2 

'9+4 
[x  +  2/  =  0. 

9  +  4       ^' 
[x  +  y  =  5. 

9^4        ^' 


4x2  + 
X  +  2/ 

X2 


y2  =  36, 
-Vl3. 


Ellipse,  semi-axes  a 
and  b  respectively 

(9x2  +  162/^=25, 
^'   l2x-3y  =  6. 


^  +y^  =  1, 

x  +  2y  =  2. 

x2  +  4  2/2  =  4, 
X  -  y  =  3. 


12x2  +  3^2  = 
|3x  +  2/  =  2. 

,9x2+42/2  = 
[x-y  =  1. 


12, 


[x  +  y  =  VI3. 

10.  Determine  k  so  that  the  line  x  — 
a;2  +  4  2/2  =  4. 

11.  Determine  m  so  that  the    line 
4x2  +  92/2  =36. 


k  shall  be  tangent  to  the  ellipse 
TOX  +  3  shall   touch  the  ellipse 


96.   Consider  the  equations 


x-y  =  2, 
y-  ^  ix. 


97' 


SIMULTANEOUS   QUADRATICS 


69 


Y 

/ 

v^ 

^ 

■^ 

V 

^ 

■^ 

/ 

/ 

/ 

/ 

/ 

0 

/ 

f 

/ 

X 

/ 

\ 

/ 

^^ 

■ 

^^ 

•^ 

_ 





1 

^ 

■ 

Solving  as  in  (88),  we  find 

xi  =  4  +  2  V3,  a^o  =  4  -  2  V3, 

?/i  =  2  +  2  V3,  7/2  =  2-2  V3. 

The  graphs  are  shown  in  the  figure, 
that  of  the  equation  y-  =  4  x  being  a 
parabola,  whose  vertex  is  at  the  origin 
and  whose  axis  is  the  x-axis. 

Exercise  1.  Compare  the  graphic  solution 
with  tliat  obtained  by  formula. 

Exercise  2.     For  what   value  of  k  will   the  Parabola,  y^  ==  4  x 

line    X  —  y  =  k  be   tangent    to   the    parabola 
U-  =  4  X?     Why  arc  there  not  two  values  of  k  as  in  the  exercise  of  (95)  ? 

97.   Standard    Equations    of    the    Parabola.  —  The    equation 

2/'  =  4  ax 

always  represents  a  parabola,  whose  vertex  is  at  the  origin  and 
whose   axis  is  the  x-axis.     The  curve  extends  to  the  right  from 
0  when  a  is  positive,  to  the  left  when  a  is  negative. 
The  equation 

ic^  =  4  « J/ 

always  represents  a  parabola,  whose  vertex  is  at  the  origin  and 
whose  axis  is  the  7/-axis.  The  curve  extends  upward  when  a  is 
positive,  downward  when  a  is  negative. 

Parabolas 


\ 

Y    .r=  4  ay 

\ 

"/        \ 

/ 

\ 

1 

i 

\ 

x=  =  -4 

ay 


y-  =  —  4  ax 


y-  =  4  ax 


Exercises.     Solve  and  check  l^y  graphs: 
[y  =  X.  (X  +  y  =  \. 


;4y2=x, 
|2x-2/  =  4. 


70 


SIMULTANEOUS   QUADRATICS 


[  98,  99 


\Zx  +  y  =  3. 


\y  =  2x. 

(  .t2  =  4  y, 
\x+2y- 


:  2  X  +  5  2/  =  10. 

;x2  =  -4?/, 

!2/-2x  =  l. 


10.    Determine  k  so  that  the  line  3  x  +  y  =  A;  shall  touch  the  parabola 
2/2 +4x  =0. 

11.  Determine  m  so  that  the 
line  y  =  mx  +  2  shall  touch  the 
parabola  t/2  =  8  x. 

98.    Consider  the  equations 

x-2y  =  Z, 


Hyperbola,  '^^  -  ^j  =  1 
Straight  Line,  x—  2y  =  Z 


The  graphs    are   shown   in 
the  figure. 

The  graph  of  the  second 
equation  is  an  h3rperbola,  a 
curve  consisting  of  two  open  branches  which  continually  ap- 
proach the  diagonals,  produced,  of  the  dotted  rectangle,  but  never 
cross  them;  These  lines  are  called  the  asymptotes  of  the  hyper- 
bola.    0  is  the  center  and  A' A  the  axis  of  the  curve. 

Exercise.     Compare  the  solution  of  given  equation  as  obtained  by  formula 
with  that  from  the  graph. 

99.    Standard    Equation    of    the    Hyperbola.  —  The    equation 

^3         ^.2 


b" 


=  1 


always  represents  an  hyperbola  whose  axis  coincides  with  the 
X-axis,  and  whose  center  is  at  the  origin.  The  curve  lies  between 
its  asymptotes,  which  are  the  diagonals,  produced,  of  a  rectangle 
whose  sides  are  2  a  and  2  b,  parallel  to  the  coordinate  axes,-  with 
its  center  at  the  origin. 
The  equation 

^  _y^_  ^  _  ^ 

a^       b' 


represents  an  hyperbola  whose  axis  coincides  with  the  y-axis. 


100,101]  SIMULTANEOUS   QUADRATICS 


71 


Hyperbola,  -^  —  vj  =  1 


a'       h 
Exercises,     Solve  and  check  by  graphs: 

4.  ^  9  ~  4 

5x  +  2/ 


1. 


2. 


\2x 


-y- 

32/ 


2/" 


(X2-2/2 


^  =36, 


6. 


2/ =  2. 


erbola, 

a?       ¥ 

7. 

2  x2  -  3  y=  =  6, 
3x  +  y  =  6. 

8. 

(.r2-2/2=_i, 
l2/-3x  =  l. 

9. 

(4 x2  -  9 2/2  =  - 
122/ -x=0. 

36, 

fc  shall  be  tangent   to 

the 

5. 
;4x2  -9  2/2  =36, 

|4x  +  2/  =  2. 
[x2  -4?/2  =4, 
12/  =  2x-6. 

10.  Determine  k  so  that  the  hne  x  —  2y  = 
hyperbola  4  x2  -  9  2/2  =  36. 

11.  Determine  m  so  that  the  line  y  =  mx  -  2  shall  touch  the  hyperbola 
x2  -  2/2  =  1. 

100.   Rectangular  Hjrperbola. —  The  equation 

always  represents  an  hyperbola  whose  asymptotes  are  the  coordi- 
nate axes;  for  the  upper  sign,  its  branches  lie  in  the  first  and  third 
quadrants,  and  for  the  lower  sign  in  the  second  and  fourth  quad- 
rants. \  Reclanqular 
hyper 


Red.  Hyp.         xy  =  ¥  Red.  Hyp.         xy  =  -  1^ 

101.   The  general  equation  of  the  second  degree, 

«x-  +  by'  +  cxy  -\-  dx -\- ey  +f  =  0, 

includes  all  the  types  of  equation  considered  in  the  preceding 

sections  and  always  represents  one  of  the  curves  there  shown, 


72  SIMULTANEOUS   QUADRATICS  [102 

except  in  isolated  eases  when  it  can  be  factored  into  linear  fac- 
tors, in  which  case  it  represents  a  pair  of  straight  lines,  or  when 
it  is  satisfied  by  the  coordinates  of  a  single  point  only,  as 
x2  ^  y2  =  0.  The  graph  may  also  be  imaginary,  that  is,  the 
equation  cannot  be  satisfied  by  any  real  values  of  x  and  y, 
as  a;2  -f-  i/2  =  —  1. 

The  curves  represented  by  the  general  equation  of  the  second 
■degree  are  not  restricted  in  position  with  respect  to  the  coordi- 
nate axes  as  are  those  shown  in  the  preceding  figures.  The 
center,  vertices,  axes  and  asymptotes  may  have  any  position 
whatever,  depending  on  the  numerical  values  of  the  coefficients 
a,  6,  c,  d,  e. 

All  curves  represented  by  equations  of  the  second  degree  in 
X  and  y  may  be  obtained  as  plane  sections  of  a  circular  cone.  They 
are  therefore  called  conic  sections. 

102.  Exercises.  Give  what  facts  you  can  about  the  curves 
represented  by  the  following  equations,  without  drawing  the 
graphs : 

1.  a;2  +  2/2  =  9.  11.   x2  =  4  y. 

2.  4  x2  +  4  2/2  =  16.  12.   4  x2  =  2/. 

3.  3x2+3  2/2  =  15. 

4.  4  x2  +  2/2  =  4. 

5.  x2  +  4  2/2  =  4. 

6.  16x2+25  2/2  =  400. 

7.  25x2  +  16  2/2  =400.  » 

8.  2x2+4  2/2  =  9. 

9.  2/2=4  X. 
10.   4  2/2  =  X. 

Construct  the  graphs  of  the  preceding  £ 
Construct  the  graphs  of  the  equations: 

21.  x2  +  2/2  -  6  X  -  8  2/  =  0. 

22.  (X  -  2/)2  =  1. 

23.  3x2+2x2/  +  32/2-162/  +  23=0. 

24.  x2  -  5  xy  +  6  2/2  =  0. 

25.  3  x2  +  2  2/2  -  2  X  +  2/  -  1  =  0. 

Solve  graphically  and  by  formula  several  of  the  preceding  equations  with 
the  equation 

(a)  x-2/  =  1.  (b)  2X  +  3  2/  =  6. 

(c)  x  +  2/  =  0.  (d)  2x-2/  =2. 


13. 

2/2=  -4x. 

14. 

-  4  2/2  =  X. 

15. 

x2  =  -  4  y. 

16. 

4x2   =_y. 

17. 

16  x2  -  25  2/2  =  400. 

18. 

16x2  -25  2/2  =-400. 

19. 

25x2-16  2/2  =  400. 

20. 

25  .x2  -  16  2/2  =  -  400. 

qual 

tions  on  cross-section  papei 

26. 

x2  +  2  X2/  +  2/2  =  0. 

27. 

5  x2  +  2  X2/  +  5  2/2  =  0. 

28. 

4.r2/+6x  -S2/  +  I  =0. 

29. 

2/2  -  X2/  -  5  X  +  5  2/  =  0. 

30. 

xy  -2/2  =  1. 

103-105]  SIMULTANEOUS  QUADRATICS  73 

103.  Solution  of  Two  Simultaneous  Quadratics.  —  When  both 
quadratics  are  of  tlie  general  form,  as 

ax~  +  by-  -\-  cxy  -\-  dx  -\-  cy  -\-  f  =  0, 
a'x^  +  b'y'~  +  c'xy  +  d'x  +  c'y  +/'  =  0, 

they  cannot  usually  be  solved  by  elementary  methods.  For,  if 
we  solve  one  equation  fOr  y  in  terms  of  x  say,  and  substitute  in 
the  other,  we  obtain,  after  rationalizing,  an  equation  of  the  fourth- 
degree  in  X.  Such  an  equation  requires  rather  complicated  pro- 
cesses for  its  solution.  We  shall  therefore  leave  aside  the  general 
case  and  discuss  some  special  cases,  such  as  usually  arise  in  the 
practical  application  of  algebra.  We  begin  with  some  graphic 
illustrations. 

104.  Graphic  Solution.  —  Since  each  of  the  above  equations 
represents  graphically  a  conic  section,  two  such  curves  intersect 
in  general  in  four  points.  All  real  solutions  are  shown  by  the 
intersections  of  the  graphs,  and  may  be  read  off,  approximately 
at  least,  from  the  diagram. 

Whe7i  the  graphs  intersect  in  less  than  four  points  (tangency  is 
counted  as  two  coincident  points  of  intersection),  some  solutions  are 
imaginary  or  infinite. 

The  various  cases  which  may  arise  are  illustrated  in  the  figures 
on  page  74. 

We  proceed  to  consider  some  special  cases  of  simultaneous 
quadratic  equations. 

105.  Case  1.    Two  quadratics,  one  of  which  is  factorable. 
Ride:  Factor  the  equation,  put  each  factor  equal  to  zero,  and 

solve  each  of  the  resulting  linear  equations  with  the  other  quad- 
ratic. 

Rule  for  factoring  a  quadratic.  Solve  for  y  in  terms  of  x  (or  x 
in  terms  of  y);  if  the  quantity  under  the  radical  is  a  perfect 
square  the  two  values  of  y  are  of  the  form  y  =  ax  -{-  b  and 
y  =  a'x  +  6.     The  required  factors  are  then 

(y  -  ax  -b)  (y  -  a'x  -  b'). 

Graphically,  the  factorable  quadratic  represents  a  pair  of 
straight  lines,  the  other  quadratic  some  conic.  Each  straight 
line  may  cut  this  conic  in  two  real  distinct  points,  in  two  real 
coincident  points,  or  in  two  imaginary  points  (i.e.  does  not  cut  at 


74 


SIMULTANEOUS   QUADRATICS 


[105 


1  +  ^  =  1  E_  +  y  =  1 

9       4  9^4 

Four  real  solutions,  Four  real  solutions, 

all  distinct.  two  being  equal. 


t  +  t  =  l 
9^4 

(x  -  iy+  if  =  y 

Two  real  distinct 

solutions,  two 

imaginary. 


Two  real  and  equal  solutions, 
two  imaginary. 


X'  -  7/  =  1 

X  -  y   =Q 

Two  solutions,  both  infinite. 


&Q 


f-|'  =  l;i(x-6)^  +  r  =  f 
All  four  solutions  imaginary. 


xy  =  1 
xy  =  -  I 
Four  solutions,  all  infinite. 


The  student  is  urged  to  draw,  or  to  picture  to  himself  mentally  as 
far  as  possible,  graphs  corresponding  to  all  equations  considered. 
He  should  be  able  to  recognize  at  a  glance  the  standard  forms  of 
equation  of  the  conic  sections. 


105] 


SIMULTANEOUS   (QUADRATICS 


75 


all).     Hence  the  four  solutions  may  be  all  real  and  distinct,  or 

equal    in  pairs,    or    imaginary    in 

pairs. 

x2-2.Ti/-3i/2  =0, 

x2  -  4  (/2  -  4  =  0. 


Example  1. 


The  factors  of  the  first  equation  are,  by 
inspection, 

{x  +  7j)ix  -3y)  =  0. 
X  +  y  =  0     or     X  -  3  2/  =  0. 


\ 

Y 

^. 

\ 

./ 

>s 

\ 

^> 

\  N 

o_. 

^ 

f 

X 

r 

\ 

\ 

^ 

^ 

\ 

xj 

^ 

\ 

\ 

\ 

Hence  we  have  to  solve 


Hyperbola,  x^  —  Ay-  —  4  =0 
Straight  lines,  x^  —  2xy  —  3  ?/'  =  0 
or        X  -V  y  =  0  and  x  —  3  y  =  0 


U  +  2/  =  0, 
( x2  -  4  y2  _  4 


0, 


Solving  the  first  pair,  we  have 


{^h  y\) 


h) 


and 


(^2,  2/2 


3  2/  =0, 

-42/2-4 


4  4 


These  are  imaginary.     The  line  x  +  ?/  =  0  does  not  cut  the  hyperbola  (figure). 
Solving  the  second  pair, 


fe,  2/3) 


V5  Vo/'     '        V    Vs 


These  solutions  are  real,  and  the  approximate  values  may  be  scaled  off  from 
the  figure. 

Note.     An  equation  of  the  form   Ajc'  +  Bxij  +  Cj/"'  =  0   cayi 
always  he  factored.     Divide  by  the  square  of  one  of  the  variables, 

and  solve  for  the  ratio  -  or  -• 
X       ij 

The  factors  will  be  imaginary  if  5-  —  4  ^C  <  0,  and  in  this 

case  the  graph  of  the  equation  is  imaginary.     In  all  other  cases 

the  graph  is  a  pair  of  real  straight  lines,  distinct  if  5^  —  4  AC  >  0, 

and  coincident  if  .B-  —  4  ^C  =  0. 

Example  2.     Factor  2  x2  -  2  xi/  +  ?/2  =  0. 
Divide  by  x2:  /^ V _  2  ^|  j  +  2  =  0. 

^  =  1  +  V^n[  or  1  -  V^l. 

X 

Hence  the  factors  are 

[2/  -  (1  +  V^T)  x]  [2/  -  (1  -  V^^)  x]  =  0. 


76 


Example  3. 


SIMULTANEOUS   QUADRATICS 
0. 


[106 


i2x2-2/2 
\x^  -Ay 


-  xy  +  3  2/  -  2 
0. 


\ 

\ 

Y 

20 

/ 

\ 

0 

/ 

V 

10 

/. 

/ 

\ 

■'          / 

1/ 

v^ 

/ 

^ 

K 

!■ 

y 

A 

i 

\IO 

Solving  the  first  equation  for  x  in  terms  of 
we  have 


Parabola,         a;^  -  4  y  =  C 
Straight  lines, 

2x'-y--xij  +  3y-2 
or  x  —  y  +  1  =  0 

and        2x  +  y-2==0 


?/±  V9?/2-24y  +  16^2/±  (3y-4) 
4  4 


Hence, 

X  -y  +  1  =0    or    2x+?/-2=0. 

Solving   the  first   of  these   with  the  second 
equation  above,  we  have 

ixi,yi)  =  (2+2V2,  3+2V2); 
(0:2,2/2)  =  (2-2V2,3-2V2). 

From  the  second  equation  we  obtain 
(x3,2/3)  =(-4  +  2V6,  10-4V6); 

(.T2, 2/2)  =  ( -  4  -  2  V6, 10  +  4  Ve). 

Exercises.     Solve  for  x  and  y,  and  check 


graphically: 


(x2 +  2/2  =  1, 

\x^  +  yx  -2  2/2=0. 

(3:2  +  2/2  =4, 

^^2-  2/2=0. 

( 4  x2  +  9  2/2  =  36, 

(2x2 +  5x2/ +32/2  =6x  +  62/. 


(X2-2/2  =  1, 

|x2/-22/+x  =2. 

( 2/2  -  4  X  =  0, 
|6x2+x2/- 12  2/2  = 


4  2/2 


106.   Case  2.     Homogeneous  equations. 

Definition.  An  equation  is  called  homogeneous  when  all  of  its 
variable  terms  are  of  the  same  degree.  A  constant  term  may  be 
present.  (In  the  further  developments  of  mathematics,  the  last 
sentence  is  omitted  from  the  definition.) 

Two  homogeneous  quadratics  have  the  forms 

Ax^  +  Bxij  +  Ci/  =  D, 


(1) 


D'. 


(2)  AV  +  B'xy  +  CY 

Solution.     Multiply  the  first  equation  by  D',  the  second  by  D 
and  subtract.     The  result  is  a  new  equation  of  the  form 

(3)  A"a;2  +  B"xy  +  C'Y~  =  0, 

which  may  be  solved  with  either  of  the  given  equations  by  factoring, 
as  in  Case  1. 


106] 


SIMULTANEOUS   QUADRATICS 


77 


Graphically,  equations  (1)  and  (2)  represent  two  conies,  and 
equation  (3)  a  third  conic  which  consists  of  a  pair  of  straight  lines 
in  case  the  factors  are  real.  Conic  (3)  goes  through  the  inter- 
sections of  (1)  and  (2),  since  the  coordinates  of  any  point  which 
satisfy  (1)  and  (2)  will  also  satisfy  (3).  Hence,  when  the  factors 
of  (3)  are  real,  we  obtain  the  intersections  of  (1)  and  (2)  by  finding 
the  intersections  of  either  of  them  with  a  pair  of  real  straight  lines. 
When  these  factors  are  distinct,  there  are  two  distinct  lines,  either 
of  which  may  cut  the  conic  in  two  real  and  distinct  points,  two 
coincident  points,  or  two  imaginary  points.  When  the  factors  are 
imaginary  the  lines  are  imaginary,  and  all  four  solutions  are 
imaginary. 

Another  method  of  solving  two  homogeneous  equations  in  the 
forms  (1)  and  (2)  is  to  put  in  both  of  them  y  =  vx.  Then  divide 
one  equation  by  the  other,  and  clear  of  fractions,  after  removing 
the  common  factor  x^.  The  result  is  a  quadratic  in  v,  whose  roots 
we  may  represent  by  vi  and  v-z.  Then 
y  =  v\X   and   y  =  vox. 

Substituting  these  values  in  turn 
in  either  of  the  given  equations, 
we  have  two  quadratic  equations 
in  X  alone. 


Example  1. 


2x2 
4x2/ 


3  x;/  +  4  =0, 
5  7/2-3=0. 


Transposing  the  constant  terms  we  have 
2  x2  -  3  x?/  =  -  4. 

4  x?/  -  5  2/2  =  3. 

Multiplying  the  first  equation  by  3,  the 
second  by  4,  and  adding, 

6  x2  +  7  xy  -  20  2/2  =  0 
or         (3  X  -  4  2/)  (2  X  +  5  2/)  =  0. 


— 

— 

- 

"f 

- 

/I- 

~ 

/ 

> 

V 

^. 

/ 

'-. 

^^ 

'/ 

(Z) 

'TzT 

"" 

~~- 

,' 

'  ■■- 

X 

/ 

" 

~- 

/. 

>^ 

\ 

" 

^ 

^ 

„        ,,       i2x'-3x2/  +  4  =  0 
3  X  -  .4  y  =  0 


Straight  lines, 


\2x  +  5y 


Equating  each  of  these  factors  to  zero,  and  solving  with  one  of  the  given 
equations,  we  have,  from  the  first  factor, 


(xi,  y\) 
from  the  second  factor, 

N,  2/3)  =  (2  V^  - 


(4,3);  (x2,2/2)  =  (-4,  -3); 


J  V^);  (x4,2/4)  =  (-  h  V-5,  I  V-5). 


Hence  two  solutions  are  real  and  two  imaginary.      The  figure  shows  the  graphs 
of  the  given  equations  and  of  the  factors  of  the  auxiliary  equation. 


78 


SIMULTANEOUS   QUADRATICS 


[107 


To  solve  by  the  second  method,  transpose  the  constant  term  as  before, 
then  put  y  =  vx. 

4  !;x2  -  5  !;2x2  =  3. 
2-3?;  4 


4  V  —  5  1^2 


Clearing,  etc.,    20v'^ -7  v  -  &  =  Q. 
Hence,  ?;  =  f  or  —  f . 

Therefore  y  =  Ix  or  y  =  —  Ix. 

(These  are  the  linear  factors  of  the 
auxiliary  equation  found  above.) 

Substituting  these  values  of  y  in 
either  of  the  given  equations,  we  find 
X  as  before. 
Example  2. 

9  x2  +  a;!/  +  2  ?/2  =  60, 
8  x2  _  3  a;?/  -  2/2  =  40. 
The  auxiliary  equation  is 

6  x2  -  11  x?/  -  7  2/2  =  0, 
or       (2  X  +  ?/)  (3  X  -  7  2/)  =  0. 

Solving  each  factor  with  one  of  the 
given  equations  we  obtain 
(xi,  2/i)  =  (2,  -4);  (X2,  2/2)  =  (-2,4); 


Ellipse,  9  x=  +  xy  +  2  2/2  =  60 

Hyperbola,  Sx^-  Zxy  -y-  =  40 

Straight  lines,  {2x  +  y)  {3x  -  7 y)  =  0 

The  graphs  are  given  in  the  figure. 
Exercises.     Solve  for  x  and  y: 


(^^'^^^=(^'^)  = 


(x2  +  2/2    =9, 
I  x2  -  X2/  =  10. 

JX2-2/2   =   1, 

(x^  —  xy  +  y^  = 


( 4  x2  -  9  2/2 
\y^  +  xy  = 


{xi,  2/4) 


7__ 
V2' 


_3 

2V2 


}■ 


(x2+2x2/  =  2, 
1.        *•   (2x2/ -2/'  =6. 


(  x2  +  X2/  +  2/2  =  3, 
1 2  x2  -  3  2/-  =  6. 

i2.r2  +  X2/-3  2/2  =  2, 
\x'--xy  +  2y^  =  l. 


107.   Case  3.     The  given  equations  are  of  the  forms 
ax-  +  by^  =  c, 


Rule.     Consider  a;^  and  \f-  as  the  unknowns,  and  solve  by  the 
method  of  Hnear  equations. 


lOS] 


SIMULTANEOUS   QUADRATICS 


79 


Graphically,  we  have  two  conies  in  standard  form.     The  four 
solutions  may  all  be  real,  or  equal  or  imaginary  in  pairs. 

Example,    x-  —  4  y^  =  4, 

9  x2  +  16  2/2  =  144. 
By  elimination  we  obtain, 

Hence  x  =  ±  4  Vi§; 

2/  =  ±  3  VS. 
Taking  eitlier  value  of  x  with  either 
value  of  y,  we  obtain  the  four  solutions. 
The  approximate  values  may  be  scaled 
off  from  the  Figure. 


r 

>^  ^-^ 

"~~~   V-                          ^ 

-  ^^ 

\'^  ' 

^ 

o     7J\        x 

I  2 

\i 

X 

>^    ' 

^      ^-^ 

--^  ^: 

Hyperbola,  x^  —  4  t/^  =  4 
Ellipse,  9x'  +  16?/'  =  144 


Exercises.     Solve  for  x  and  y,  and  check  graphically: 
^    (x2  +  ;/2=4,  3_   j2x2  +  52/2  =  10,  ^ 


1x2-7/2=2. 
(X2-  2/2   =   1, 

1x2  +  4  2/2  =  4. 


I  4  x2  +  2/-  =  4. 

:x2  +  2/2  =  9, 

'  4  x2  +  9  2/2  =  36. 


( x2  -  2/2  =  9. 

i  x2  +  2/-  =  1, 
1x2  +  2/2  =4. 


108.   Case  4.     Symmetric  and  Skew-Symmetric  Equations. — 

A  symmetric  equation  is  one  which  remains  unchanged  when  the 
variables  are  interchanged. 

A  skew-sijmmetric  equation  is  one  whose  variable  terms  all  change 
sign  when  the  variables  are  interchanged.     Thus 

x^  +  y^  +  X  +  y  =  0,         x'^-y^  +  2x-2y=\ 

are  symmetric  and  skew-symmetric  respectively. 
Rule.     Given  two  such  equations,  put 

x  =  u  -{-  V     and     y  =  u  —  v; 
solve  the  resulting  equations  for  u  and  v;  then 

X  =  ^  (u  -{-  v)     and     ?/  =  V  (m  —  v). 

Note.     Equations  of  higher  degree  than  the  second  may  often 
be  solved  by  this  methorl. 


Example. 


x^  +  y' 
x~  +  y 


:-y-  =  9, 
■vy  =  3. 


Let 


u  +  V     and     y 


80  SIMULTANEOUS   QUADRATICS  [109 

"Substituting  and  reducing: 

u*  +  14  w2y2  +  ?;4  =  9, 

m2  +  3  y2  =  3. 

Let  u2  =  s     and     v"^  =  t. 

Then  s^  +  U  si  +  t^  =  9, 

s  +  3  <  =  3. 
Solving:  (s,  <)  =  (3,  0)or  (f,  I). 

(If  s  and  /  be  considered  as  the  coordinates  of  a  point,  the  equations  in 
s  and  t  represent  an  ellipse  and  a  straight  line  respectively.) 

Since  w  =  ±  Vs     and     v  =±^l, 

we  have  (m,  f)  =  ( ±  Vs,  o)    or     (±^'    ±^ 

where  the  signs  are  to  be  taken  in  all  possible  ways. 
Then 

X  =  w+t;  =  VS, -V3,  V3, -V3,       0,       0; 

y  =u-v  =  -\JZ,  -\l2,,      0,         0,     V3,  -  V3. 
Here  corresponding  values  of  x  and  y  appear  in  the  same  vertical  line. 

109.  Case  5.  Symmetric  Solution.  —  This  method  of  solution 
is  applicable  to  certain  forms  of  symmetric  equations,  and  may  be 
illustrated  by  some  simple  examples. 

Example  1.  x  -\- y  =  5, 

xy  =  4. 
Squaring  the  first  equation:  x"^  -\- 2  xy  +  y"^  =  25. 

Subtracting  four  times  the  second :  x2  —  2  x?/  +  2/2  =  9. 
Hence  x  -  y  =  ±  Z.  .     ' 

But  X  +  2/  =  5. 

X  =  4  or  1 ;     y  =  1  or  4. 

Example  2.  (1)     x^  +  xy  +  y^  =  6. 

(2)     x2  -  xy  +  2/2  =  10. 
Subtract  (2)  from  (1) :         2xy  =-  4,     or     xy  -' -  2. 
Add  x2/=-2to(l):       x2  +  2x2/ +  2/2  =  4,       or    x  +  y=±2. 

Subtract  3  x?/  =  -  6  from  (1) :   x2  -  2  X2/  +  2/^  =  12,     or    x  -y  =±2  V3. 
Hence  x  =  ±  1  ±  Vs     and     2/  =  ±  1  T  \/3. 

Simultaneous  values  of  x  and  y  are  then  obtained  by  taking  the  same  com- 
bination of  signs  in  these  two  results. 


110]  SIMULTANEOUS  QUADRATICS  81 

110.  Miscellaneous  methods  for  solving  two  simultaneous 
equations. 

These  methods  depend  on  reducing  the  given  equations,  which 
may  be  of  higher  degree  than  the  second,  to  one  of  the  cases 
already  discussed. 

1.  By  Substitution.  —  This  method  has  already  been  illustrated 
in  several  cases;  in  (106)  we  made  the  substitution  y  =  vx,  in  (107) 
we  put  X  =  u  -\-  V  and  y  =  u  —  v,  and  in  example  2  of  (107)  we 
put  u-  =  s  and  v^  =  t.  We  shall  give  two  more  simple  illustra- 
tions. 

1 


Example  1. 

i-.-15. 

xy 

If  we  let  ^  =  s 

X 

and 

1  _ 

y  ~ 

t,  and  we  obtain, 
s  +  <  =  2, 
St  =-  15. 

These  may  be  solved  by  the  method  of  (109). 

Example  2. 

+  x^y2  +2xy  =  4, 
x2i/2  -2xy  =0. 
Let        X  -{-  y  =  s  and  xy  =  t.     Then 

<2  -  2  <  =  0. 
The  last  two  equations  are  readily  solved,  and  give 
s  =  +  2;   -  2;  0. 
^  =       0;        0;  2. 
The  values  of  x  and  y  may  now  be  found  by  solving  the  pairs  of  equations, 
(x+2/  =  2,  (x  +  y=-2,  (x  +  i/  =  0, 

(xy  =0.  \xy  =0.  \xy  =  2. 

2.  By  modifying  or  combining  the  given  equations  so  as  to 
obtain  simpler  forms.  In  particular,  a  common  factor  may  some- 
times be  removed  by  division. 

Example  1. 

(1)  x^  -xy  =  18  y, 

(2)  xy  -y^  =2  x. 
Dividing  (1)  by  (2),  we  have 

-  =  9^     or    (^y=  9     or    x  =  ±  3  y. 
y         X  \yj 


82  SIMULTANEOUS  QUADRATICS  [111 

Substituting  each  of  these  values  of  x  in  either  of  the  given  equations,  we 
can  solve  for  y  and  so  complete  the  solution. 
Example  2. 

(1)  ix'^y-x  =  l,  {x(x2/-l)=l, 

(2)  ( a:%2  -x^-  =  3;       \  x'~  (x%2  _  1)  =  3. 

Divide  (2)  by  (1):  x  (xy  +  1)  =  B. 

Divide  this  equation  by  (1):  ^    ^  ^  =  3. 

Hence  xy  =  2. 

Then  from  (2),  x^  (4  -  1)  =  3,,    or     x^  =  1,     or     x  =  ±  1. 

But  from  (1),  x  (2  -  1)  =  1,     or    x   =  1. 

In  this  case  the  value  x  =  —  1  must  be  discarded. 
Hence  the  only  solution  is  x  =  1,  y  =  2. 
Examiple  3. 

(1)  x4  +  y"  =  1, 

(2)  X  -  y  =  1. 

Raise  (2)  to  the  fourth  power  and  subtract  from  (1): 

(3)  4  x3y  -  6  x2y2  +  4  xy3  =  0. 
Square  (2)  and  multiply  the  result  by  4  xy: 

(4)  4  x3y  -  8  x2y2  +  4  xy^  =  4  xy. 

Subtract  (4)  from  (3) : 

2  x2y2  =  -  4  xy,     or     x2y2  +  2  xy  =  0. 

Hence  xy  =  0,     or     xy  =  -  2. 

Solving  each  of  the  last  two  equations  with  (2)  we  have 

(x,y)  =  (l,0),  (0,-l),(^^^  ,  V^^j\—^'  2 }' 

All  four  solutions  also  satisfy  equation  (1). 

111.  Summary  of  Methods  for  Solving  Simultaneous  Equa- 
tions.—  [Let  the  given  equations  be  numbered  (1)  and  (2).] 

(a)  Equation  (1)  linear,  (2)  quadratic. 

Rule:   Substitute  from  (1)  in  (2).     Graph,  straight  line  and  conic. 

(b)  Equations  (1)  and  (2)  both  quadratic. 
Case  1.    Equation  (1)  is  factorable. 

Rule:  Put  each  factor  separately  equal  to  zero  and  solve  with 
(2)  as  in  (a).     Graph,  two  straight  lines  and  a  conic. 

Rule  for  factoring:  Solve  for  y  in  terms  of  x  (or  x  in  terms  of  y) ; 
the  quantity  under  the  radical  must  be  a  perfect  square. 


Ill]  SIMULTANEOUS  QUADRATICS  83 

Case  2.  (l)  Ax' +  Bxij -\- Cif=  D;  (2)  A'x' -{- B'xy +  CY  =  D\ 
Form  the  auxiliary  equation,  (1)  X  D'  —  (2)  X  Z>  =  0.  Factor 
this  and  solve  as  in  Case  1. 

Second  Method:  Put  y  =  vx  in  (1)  and  (2)  and  divide  results. 
Graph,  two  conies,  centers  at  origin  (except  in  case  of  parabola.) 

Case  3.     (1)  Ax^  +  By^-  =  C;  (2)  A'x'  +  BY  =  C. 

Solve  as  linear  equations  for  x^  and  y~. 

Graph,  two  conies  in  standard  position. 

Case  4.    Symmetric  Equations. 

Put  X  =  u  -\-  V  and  y  =  u  —  v. 

Applicable  to  equations  of  higher  degree. 

Case  5.    Symmetric  Solution  of  certain  symmetric  equations. 

(c)  Miscellaneous  Methods. 

Exercises. 


1. 

x2  +  y2  =  661. 

8. 

x  +  y     j<, 

15. 

5  X  +  2  2/  =  29. 

X^  -  y2  =  589. 

X  -y 

5  X2/  =  -  105. 

2. 

2/2  -  x2  =  -  80. 

x^-y^=  48. 

16. 

xy  =  80. 

x'-  +  f'  =  82. 

9. 

5  x2  +  2  2/2  =  373. 

x  =  52/. 

3. 

3x2-^2  =  59. 

2x  +  5  2/  =  54. 

17. 

4x2-3  2/2  =  -83. 

2a;2  +  3  7/2  =  98. 

10. 

x2  +  2/2  =  10. 
X  -  2/  =  2. 

3x  +  22/  =  26. 

4. 

x  +  2/  =  12. 
X2/  =  35. 

11. 

x2  -  2/2  =  120. 
X  +  2/  =  20. 

18. 

3x2-2/2  =  83. 
X  +  2/  =  15. 

5. 

x  +  y  =  1. 
xy=  -I. 

12. 

x2  -  2/2  =  -  i 
x  +  y  =  l 

19. 

X2/  +  X  =  20. 
xy  -  y  =  12. 

6. 

x^  +  y^  =  74. 
x  +  y  =  12. 

13. 

x2  +  X2/  =  260. 
xy  +  y^  =  140. 

20. 

2  X  +  3  2/  =  20. 
Sxy-y'~  =38. 

7. 

x  +  y  =  l 
xy  =  h 

14. 

x2  +  2/2  =  218. 
xy-y^=  42. 

21. 

5x2-42/2  =  109. 
7  X  -  5  2/  =  25. 

22. 

x+xy  +  y  =  4:7. 

27. 

x2  +  X2/  +  2/2  =  4. 

x  +  y  =  12. 

x2 

-  xy 

'+2/- =2. 

23. 

x2  +  xy  +  2/-  =  217. 

28. 

X2 

+  xy  +  2/2=7. 

x  +  y  =  17. 

X  - 

f  2/  +  xy  =  5. 

24. 

1+1=1. 

29. 

X2 

xy 

+  2/- 
=  3. 

=  5  (X  +  2/). 

1+1=1. 

30. 

.r3  +  2/3 

=  9. 

x2  ^  2/-       -i^ 

X2/ 

=  4. 

26. 

2x2 -3x2/ +  2/2  = 

3. 

31. 

X2 

-42/2=4. 

x2 +2x2/ -32/2  = 

=  5. 

X2 

-2x2/  +  2x  =42/. 

26. 

x2  -  X2/  +  2/2  =  37 

. 

32. 

2x 

:2-2 

!  2/2 +3x2/  =  -x-2j/. 

x2  -  2/2  =  40. 

X2 

-4i 

y2  -  X  +  2  2/  =  0. 

84 


SIMULTANEOUS   QUADRATICS 


33.  -u2  +  y2  -I-  uv  =  67. 
u+v  =  9. 

34.  p^  +  pq+q'^  =  79. 
p2  _  p5  +  g2  =  37. 

35.  r2+s2  +rs  =25. 
r +s  =5. 


r2  + 


-rs  ==  84. 
2. 


37.  u  +  t'  +  u2  +  i;2  =  162. 

,,  _  ,,  4-  ^2  _  i,2  =  _  102. 

38.  p+g  +  p2  +q2  =  ig. 

y  _  p  +  g2  _  p2  =    _  1. 

39.  a;2  +  2/2  +  a;  + 1/  =  18. 
2xy  =  12. 

40.  ;i2  +  fc2  _  A;  +  /i  =  32. 
2  /ifc  =  30. 

41.  x2  +  y2  +  X  +  2/  =  168. 
•Va;y  ■=  6. 

42.  m2  +  n2  -  ?rt  +  n  =  2400. 
^/mn  =  30. 

43.  9tt2+t;2+3M  +  i'  =  3042. 
Vl6w  =  48. 

44^  ^3  _  s3  =  1304. 

r  -  s  =  8. 

45.  p*  +  ^  =  337.      . 

p  +  9  =7. 

46.  x"  -  1/  =  609. 
X  -  2/  =  3. 

47.  u4  +  ?^^  =  2657. 
M+i;  =  11- 

48.  m3  +  n3  =  152. 

m2  -  mn  +  n^  =  19. 

49.  p  +  <7  +  V/H^  =  20. 
p3  +  ^3  =  1072. 

50.  .t3  +  1/  =  280. 

x2  -  X2/  +  2/2  =  28. 

51.  m2  +  3  ifl  =  7. 

7  u2  -  5  wy  =  18. 


52.  p3  -)-  gS  =  152. 
p2q  +  pg-  =  120. 

53.  x3  -  2/3  =  335. 


X2/2  -  x22/ 


70. 


54.  s3  +  i3  =  855. 

St  (s  +  0  =  840. 

55.  w3  -  n3  =  602. 
mn{n  —  m)  =  —  198. 

56.  W2y4  + 1,2  =  17. 
W2;2  +  y  =  5. 

57.  xl  -{-yi  =  35. 
xi  +  2/^  =  5. 

58.  x2|/2  -18x2/ +  72  =0. 
6x2 -17x2/ +  12  2/2  =0. 

69.  x4+x22/2+2/4  =91. 
x2  -  X2/  +  2/2  =  7. 

60.  x3  -  2/3  =  7  (x2  -  2/2). 
x2  +  2/2  =  10  (x  +  y). 

61.  s6  +  <6  =  65. 

S4  +  i4   =  17. 

62.  x2  +  2/2  =  a. 
x2  -  2/2  =  6. 

63.  X  -  2/  =  wi. 
X2/  =  n2. 

64.  7^2  +  g2  =  a2. 
p  +  g  =  6. 

65.  Vw  +  V^  =  «• 

u  + 1;  =  62. 

66.  x2  +  2/2  =  a  (x  -  y). 
x2  +  2/2  =  b  (x  +  2/). 

67.  ax-by  =  m. 
a3x3  -  632/3  =  nx2/. 

68.  6(x  +  2/)  =a(x-2/)- 
x2  +  2/2  =  w2. 

69.  x*  +  2/*  =  -  8- 
X  -  2/  =  2. 

70.  p*  +  g^  =  -  9- 
p  -(/  =  3. 

71.  u*  +  t;4  =  175. 
M  —  V  =  5. 

72.  7-2  -|-  rs  +  s2  =  a. 
r3s  +  rs3  =  6. 


SIMULTANEOUS   QUADRATICS  85 


73.  l-i=l. 

X      y      a 

75. 

X3  +  X2/2   =  p. 

y3  -1-  x2y  =  q_ 

Ui^=i^- 

76. 

m^  —  7i3  =  a  (w  —  n). 

a;2  ^  y2      62 

7h3+  7t3  =  6  {)ii  +n). 

74,   ti2  4-  Mi;  =  m. 

77. 

r5  +  ,s5  =  3368. 

v^  +  7/y  =  n. 

r  +  s  =  8. 

78.   x(.r  +  y-2)  =  1.      80. 

xy  =  Sz. 

82.   X  (x  +  2/  -  2)  =  «• 

?/  (x  +  ?/  -  2)  =  2. 

xz  =  IS  2/ 

2/  (x  +  2/  —  2)  =6. 

z{x  +  y-z)  =  3. 

2/2    =   0  X. 

2  (x  +  2/  -  2)  =  c. 

79.   X  +  2/  +  z  =  2.          81. 

x?/  4-  X  = 

1. 

83.   X  +  2/  +  2  =  p. 

x;/  =3. 

2/2  +2/  = 

-  1. 

X2/  =  (7. 

xyz  =  6. 

xz  -\-  z  = 

3. 

xyz  =  r. 

84.   xix+y+z)^  a2. 

87. 

X2/  +  X  =  a. 

2/  (x  +  2/  +  0)  =  &2. 

2/2  +2/  =  ^• 

2  (x  +  2/  +  z)  =  c2. 

xz  -\-z  =  c. 

85.   (x  +y)(x+z)  =  4. 

88. 

x-  -\-  y-  =  17  z. 

(x  +  2/)(y+2)  =  1. 

3(x  +  2/)  =52. 

(x  +  2)  (2/  +  2)  =  16. 

X  -2/  =2. 

86.    (x  +  2/)  (x  +  2)  =  a. 

89. 

X2   +  2/-  +  22   =   iU. 

(x  +y}{y+z)  =  b. 

2/-+.J^  =  ^l. 

(x  +  2)  (2/  +  2)   =  c. 

22+X  =   i|. 

Problems. 

Oo^)y        ^^— 

1.   The  hypothenuse  of  a 

right  triangle  ia 

[Too"  ft.  long.     Find  the  other 

sides,  if  their  ratio  is  3  :  4. 

2.   The  product  of  two  numbers  is  735,  and  their  quotient  ^     Find  the 

numbers. 

3.  Find  two  factors  of  1728  whose  sura  is  84. 

4.  The  sum  of  two  numbers  is  34.  Three  times  their  product  exceeds  the 
sum  of  their  squares  by  284.     What  are  the  numbers? 

5.  The  product  of  two  numl)ers  increased  by  the  first  is  180,  increased  by 
the  second  is  176.     What  are  the  numbers? 

6.  The  product  of  two  numbers  times  their  sum  is  1820,  times  their  differ- 
ence 546.     What  are  the  numbers? 

7.  The  sum  of  the  squares  of  two  numbers  plus  the  sum  of  the  numbers 
is  686.  The  difference  of  the  squares  plus  the  difference  of  the  numbers  is  74. 
What  are  the  numbers? 

8.  The  diagonal  of  a  rectangle  is  89  ft.  long.  If  each  side  were  3  ft. 
shorter,  the  diagonal  would  be  4  ft.  shorter.     Find  the  sides. 

9.  The  diagonal  of  a  rectangle  is  65  ft.  long.  If  the  shorter  side  were 
decreased  by  17  ft.  and  the  longer  increased  by  7  ft.,  the  diagonal  would  be 
unchanged.     Find  the  sides. 

10.   The  diagonal  of  a  rectangle  is  85  ft.  long.     If  each  side  be  increased 
2  ft.  in  length,  the  area  is  increased  by  230  sq.  ft.     Find  the  sides. 


86  EXPONENTIAL  EQUATIONS  [113 

yj       11.    The  floor  area  of  two  square  rooms  is  890  sq.  ft.,  and  one  room  is  4  ft. 

larger  each  way  than  the  other.     Find  the  dimensions  of  each  room. 
(,^      12.    For  60  yards  of  cloth  B  pays  two  dollars  more  than  A  pays  for  45  yards. 

B  receives  one  yard  more  for  two  dollars  than  does  A.     How  much  does  each 

pay  per  yard? 

13.  Two  bodies  moving  around  the  circumference  of  a  circle  of  length 
1260  ft.  pass  each  other  every  157.5  seconds.  The  first  body  makes  the 
circuit  in  10  seconds  less  than  the  second.     Find  the  speed  of  each  body. 

14.  The  amount  of  a  capital  plus  interest  for  one  year  is  $22,781.  If  the 
capital  were  $200  larger  and  the  rate  of  interest  \%  larger,  the  amount  in 
one  year  would  be  $23,045.     Find  the  capital  and  rate  of  interest. 

15.  A  and  B  agree  to  do  a  piece  of  work  in  6  days  for  $45.  To  finish 
on  time,  they  hire  C  during  the  last  two  days,  and  consequently  B  gets  $2 
less  pay.  If  A  could  have  done  the  work  alone  in  12  days,  how  long  would 
it  take  B  and  C,  each  working  alone,  to  do  it? 

16.  The  quotient  of  a  number  of  two  digits  divided  by  the  product  of  the 
digits  is  3.  When  the  digits  are  interchanged,  the  new  number  is  \  of  the 
original.     What  is  the  number? 

17.  If  the  digits  of  a  two-figure  number  be  interchanged,  the  number  is 
diminished  by  18.  The  product  of  the  original  and  the  new  number  is  1008. 
What  is  the  original  number? 

18.  What  number  of  two  digits  is  5  greater  than  twice  the  product  of  its 
digits  and  4  less  than  the  sum  of  their  squares? 

19.  A  fraction  is  doubled  by  adding  6  to  its  numerator  and  taking  2  from 
its  denominator.  If  the  numerator  be  increased  and  the  denominator  de- 
creased by  3,  the  fraction  is  changed  to  its  reciprocal.     What  is  the  fraction? 

20.  A  and  B  start  at  the  same  time  from  two  points  221  miles  apart  and 
travel  towards  each  other.  A  goes  10  miles  a  day.  B  goes  as  many  miles  a 
day  as  the  number  of  days  until  they  meet  diminished  by  6.  How  far  did 
each  one  travel? 

21.  The  fore  wheel  of  a  wagon  makes  1000  revolutions  more  than  the 
hind  wheel  in  going  a  distance  of  7500  yards.  Had  the  circumference  of 
each  wheel  been  one  yard  more,  the  difference  between  the  number  of  revo- 
lutions would  have  been  625.     Find  the  circumference  of  each  wheel. 

22.  Find  two  numbers  such  that  their  sum  shall  be  equal  to  28,  and  the 
sum  of  their  cubes  divided  by  the  sum  of  their  squares  equal  to  1456. 

23.  Two  points,  A  on  the  x-axis  270  ft.  from  the  origin  and  B  on  the 
t/-axis  189  ft.  from  the  origin,  move  toward  the  origin.  After  10  seconds 
the  distance  between  them  is  169  ft.,  and  after  14  seconds,  109  ft.  Find  the 
speed  of  each  point. 

113.  Exponential  Equations.  —  An  exponential  equation  is  one 
in  which  the  unknown  appears  in  the  exponent.     Thus: 

Vct^  =a2^-i;  (m^+i)-^  =  ^-2^-2;  a^+i  =62x-i^ 


114]  EXPONENTIAL   EQUATIONS  87 

Exponential  equations  of  the  above  forms  may  be  solved  by 
reduction  to  ordinary  equations  by  use  of  the  principle  that 

if  a"  =  a",  then  u  =  v, 

or  more  generally, 

if  a"  =  fo",  then  w  log  a  =  t;  log  h. 

Example  1.  Vo^  =  a^^-^. 


This  may  be  written  a^  =  a^-^-i. 

•■•    1=^^-'  -  -=!■ 

Example  2. 

(TO^  +  1)x    =   ;,j-2x-2. 
m^^  +  x    =  7,1-21-2. 

x2  +  a;=-2x-2     or     x2  +  3a;  +  2 

Hence 

X  =  -  2  or  -  1. 

Example  3. 

2^+1  =  32x-i_ 

Taking  logarithms: 

(x  +  l)log2  =  (2x  -  l)log3. 

••• 

X  (log:2  -  2  log  3)  =  -  log  2  -  log  3. 

or 

log  2  +  log  3        log  6 
21og3-log2      logl' 

Using  common 

logarithms  to  four  places, 

.=  0-7781 +  _,,,,,^ 

0.6532  +       ^•^"^^''  ^• 
114.   Exercises.     Solve: 

1.  2^  =  8.  4.  (i)^  =  2i2.  7.    i0-x  =  i000. 

2.  2^  =  J.  5.  (i)x  =  1.  8.    1000^  =  100. 

3.  4x  =  Jj.  6.  {^UY  =  253.  9.   3^  +  2  =  33. 

10.  \/a^^=a-'-2.  18.  42^-3  =7x-i. 

11.  -^p2x  +  8  =  pO.  19.  a3-r+2   =  62X-1, 

12.  42*-i  =26^+8.  20.  3^'-x-6  =  1. 

13.  \frn^  =Vw^^  +  2.  21.  8^^+2-^=512. 

1  3, •  ?^  x''-! 

1*-   osi  =  Va6-i3x.  22.  5-^-2  =252'x  +  i'. 

15.  ''^■'-\/^=''"v'a"S.  23.  (ax-2)3x  +  4  =  ax(3x  +  i,. 

16.  3x  =  Vs.  24.  (6x  +  3)3x-i  =  53x(x  +  i.. 

17.  3^  +  1  =  522=.  25.  </i2^  -s/e^^  VisT^^i  =  1. 


CHAPTER  VI 

Ratio,  Proportion,  Variation 

115.  Definitions.  The  ratio  of  two  quantities  is  their  indicated 
quotient. 

Thus  the  ratio  of  a  to  6  is  r,  or  as  it  is  usually  written,  a  :  h. 

The  numerator  of  the  fraction,  or  the  first  term  of  the  ratio,  is 
called  the  antecedent,  the  other  term  the  consequent. 

The  ratio  6  :  a  is  called  the  inverse  of  the  ratio  a  :  h. 

Two  ratios  are  equal  when  the  fractions  representing  them  are 
equal. 

a      ma  ,  , 

Since  r  =  — t'    •  •  a  -.0  =  ma  :  mo. 

0      mo 

Hence,  both  terms  of  a  ratio  may  be  multiplied  by  the  same  (or  equal) 
quantities  without  altering  the  value  of  the  ratio. 

Similarly,  ii  m  9^  n,  then  a  :  b  7^  ma  :  nb. 

Hence,  if  the  terms  of  a  ratio  be  multiplied  by  unequal  quan- 
tities, the  value  of  the  ratio  is  changed. 

The  compound  ratio  of  a  :  6  and  c  :  d  is  ac  :  bd,  that  is,  the  ratio 
of  the  product  of  the  antecedents  to  the  product  of  the  conse- 
quents. 

In  particular  the  compound  ratio  of  o  :  6  and  a  ib,  or  a^  :  b'^,  is 
called  the  duplicate  ratio  of  a  to  6;  a^  :  b^  is  called  the  triplicate 
ratio  of  a  to  b,  and  so  on, 

A  proportion  is  an  equality  of  two  ratios.  Four  numbers  are 
in  proportion  when  the  ratio  of  two  of  them  equals  the  ratio  of 
the  other  two. 

Four  numbers  a,  b,  c,  d  are  in  proportion  if  a  :  6  =  c  :  rf  (often 
written  a  :  b  ::  c  :  d).  Here  a  and  d  are  called  the  extremes  and 
b  and  c  the  means.     Also,  d  is  called  a  fourth  proportional  t»  a,  b,  c. 

The  numbers  a,  b,  c,  d,  e,  .  .  .  are  in  continued  proportion  if 

a  -.b  =  b  :  c  =  c  :  d  =  d  :  e  •  •  •  . 


116]  RATIO,   PROPORTION,  VARIATION  89 

When  three  numbers  a,  h,  c  are  in  continued  proportion,  so 
that  a  :  b  =  b  :  c,  then  b  is  called  a  mean  proportional  between 
a  and  c. 

Since  ^  =  -  or  ac  =  b-  we  have  b  =  ±  *^lac.     Also,  c  is  called  the 
b      c 

third  proportional  to  a  and  b. 

116.   Laws  of  Proportion. 

1.  In  a  proportion,  the  product  of  the  means  equals  the  prod- 
uct of  the  extremes. 

2.  If  two  products,  each  containing  two  factors,  are  equal, 
either  pair  of  factors  may  be  taken  as  the  means,  the  other  as 
the  extremes  of  a  proportion. 

When  four  numbers  are  in  proportion  so  that  a  :  b  =  c  :  d, 
then  they  are  in  proportion 

3.  by  inversion,  or  b  :  a  =  d  :  c; 

4.  by  alternation,  or  a  :  c  =  b  :  d: 

5.  by  composition,  or  a  -{-b  :  b  =  c  -j-  d  :  d 
,  a      c       ,,         «,i       c    ,   I  a  +  b      c  +  d\.      /I 

6.  by  division,  ov  a  —  b  :  b  =  c  —  d  :  d; 

7.  by  composition  and  division,  ot  a  -\-  b  :  a  —  b  =  c  +  d  :  c  —  d. 

8.  Like  powers  (or  roots)  of  the  terms  of  a  proportion  arc  in 
proportion,  i.e., 

\i  a  -.b  =  c  -.d,  then  a^  ■.b''  =  c^  :  d^. 

„      ..a      c    ^,       a""      c""  \ 

For  if  r  =  3,  then  tt,  =  ^jr,-) 

0      d  b        d'  I 

9.  The  products  of  the  corresponding  terms  of  any  number  of 
proportions  are  in  proportion,  i.e.,  if 

a  :  b  =  c  :  d,  a'  :b'  =  c'  :  d',  a"  :  b"  =  c"  :  d",  etc., 

then      aa'a"  •  •  •  :  bb'b"  •  •  •    =  cc'c"  •  •  •  :  dd'd"  •  •  •  . 

/„     .,  a     c    a'     c'    a"     c"  ,,        aa'a"  .  .  .      cc'c"  .  .  .  \ 

For  if  T  =  3' r7  =  ^' r77=:7r/ •  •  •  ^  then  vtt^jt =,,,„, 

\  b     d   b'     d'   b       d"  bb'b"  .  .  .     dd'd"  .  .  .  / 

10.  In  a  series  of  equal  ratios,  the  sum  of  the  antecedents  is 
to  the  sum  of  the  consequents  as  any  antecedent  is  to  its  con- 
sequent, i.e., 

ai  :  a2  =  6]  :  62  =  ci  :  C2  .  •  • 

=  ai  +  61  +  ci  +  •  •  •  :  a2  +  62  +  Co  +  •  •  •  . 


90  RATIO,   PROPORTION,   VARIATION  [117,118 


Forif^^  =  ^^=^-i=- 

02           02         C2 

•  •  =  r,    then    ai  =  aor,  bi 

=  bir,    Cl  = 

=  C2r, 

Hence  (ai  +  h  +  ci  +  ■ 

•  ■  ■  )  =  r  {a2  +b2  +  C2  +   ■ 
ai  +bi+ci  +  ■   ■  ■  _  ^ 
02  +  62  +  C2  +  •    •    • 

■    ■    ),0T 

11.   More  generally,  if  the  ratios  ai  :  02,  61  :  62,  ci  :  C2 
all  equal  to  each  other,  then 

C  ^  ^  ^  ^  ^  ^1  =    ...    =  pai  +  9&1  +  ^ci  +  •  •  •  ^ 

^^-^  a2  ~  ^2       C2  pa2  +  g&2  +  ^C2  +  •  •  • 

where  p,  q,  r,  .  .  .  are  any  multipliers; 


n\  ^  =  ^  =  ^=  =  "/ai"  +  br  +  cr  +  •  •  • 

^  ^^  a2         62         C2         ■     ■    ■  Va2"  +  62'*  +  G2"+    •     •    • 

Exercise.  Prove  11  (a)  and  11  (b).  For  what  values  oi  p,  q,  r,  .  .  .  n 
will  these  reduce  to  10  ? 

117.  Example.     Solve  for  x:  =  j- 

X  —  a      a 

,  ,.  .  .  2x       c  +d 

By  composition  and  division:  —  =  — ZT^' 

c  +d 

X  =  a ;• 

c  —  a 

Exercises.     Solve  for  x,  using  the  laws  of  proportion: 

x+l ^3  g    x+a ^b 

•^'       X  2"  •       X  c" 

0        ^      -_^.  7.  a-x:x  =  p:g. 

^'  x-2  6 

„  8.  X  +  m  :  a  =  X  —  m  :  b. 

Q  -^^  ~  "J  _  2. 

2x  +  3       6'  9.  a-x:x-b  =  a:&. 

4.  3x-2:  3x  +  2=3:4.  ^^^  ^  ^+^ 

f.    x  +  l  _x  +  3  ^""x-p      6  +  x" 
**•  x-1       x-4' 

118.  Variation.  — A  variable  quantity  is  one  which  may  be 
considered  to  assume  a  number  of  values. 

A  function  of  a  variable  is  a  quantity  whose  value  depends  on 
that  of  the  variable. 

If  yhe&  function  of  a  variable  x  [indicated  by  writing  y  =  f  (x)], 
then  in  general,  as  x  varies  y  varies  with  it. 

Thus,  the  circumference  of  a  circle  depends  on  the  radius  and 
varies  with  the  radius.      Hence  the  circumference  is  a  function 


119,120]  RATIO,   PROPORTION,   VARIATION  91 

of  the  radius  [c  =  f  (r)].  The  functional  rolation  is  expressed  by 
c  =  2  Trr. 

Similarly,  the  area  of  a  circle  depends  on  the  radius  [A  =  f  (r)]. 
The  functional  relation  in  this  case  is  A  =  irr-. 

Also,  the  cost  of  a  piece  of  cloth  depends  on,  or  is  a  function 
of,  the  price  per  yard;  the  running  time  of  a  train  between  two 
stations  is  a  function  of  the  speed;  the  range  of  a  gun  is  a  func- 
tion of  the  muzzle  velocity. 

119.  Direct  Variation.  —  A  quantity  y  varies  directly  with  an- 
other quantify  x  when  their  ratio  remains  constant. 

This  is  indicated  by  writing  y  o:  x  (read  '' y  varies  directly 
as  X  "). 

If  k  denote  the  constant  value  of  the  ratio  of  y  to  x,  then 
y  oc  a;  is  exactly  equivalent  to  //  =  kx. 

The  constant  k  will  be  determined  as  soon  as  the  value  of  y 
corresponding  to  a  single  value  of  x  (other  than  a:  =  0)  is 
known. 

Graphically,  the  relation  between  y  and  x  is  represented  by  a 
straight  line  through  the  origin,  the  inclination  of  the  line  to  the 
X-axis  increasing  with  the  absolute  value  of  k.     The  line  is  com- 
pletel}^  determined  by  the  origin  (x  =  0,         ^ 
?/  =  0)  and  one  other  point. 

If  c  be  the  circumference  and  r  the 
radius  of  a  circle,  then  c  oc  r,  for  c  =  2  ivr. 
If  we  take  tt  =  V.  then  c  =  V  when  r 
=  1.     The  figure  gives  the  graph. 

Exercise.     From  the  figure  read  off  to       Horizontal  scale  =  10 
,,  ,  -i.    iu      1        i-u         r      •  times  vertical  scale 

the  nearest    unit  the   lengths   of  circum- 
ference of  circles  whose  radii  are  .15  in.,  .33  ft.,  1.27  mm.,  .87  cm. 
respectively. 

120.  Inverse  Variation.  —  When  y  varies  directly  as- >  that  is, 

1  k 

y  cc  -  or   y  =  -,  then  y  is  said  to  vary  inversely  as  x. 

When  y  varies  inversely  as  x,  this  may  be  expressed  by  writ- 
ing xy  =  k. 

Graphically,  the  relation  between  x  and  y  is  then  represented 
by  a  rectangular  hyperbola,  whose  asymptotes  are  the  coordi- 
nate axes. 


92  RATIO,   PROPORTION,    VARIATION  [121,122 

If  t  be  the  time,  in  hours,  required  by  a  train  to  run  10  miles, 
and  5  the  speed  in  miles  per  hour,  then 

,10  ,1 

t  =  —      or     /  cc  -  ■ 
s  s 

The  figure  gives  the  graph,  only  posi- 
tive values  being  considered. 

Exercise  1.     From  the  figure  read  off  to 

tenths  of  a  unit  the  times  required  to  run 

,      .n  10  miles  when  s  =  4.5,  7.8,  and  15.6  miles 

st  =  IQ  .  '  ' 

per  hour  respectively. 
Exercise  2.     Construct  a  curve  showing  the  possible  dimen- 
sions of  a  rectangle  whose  area  must  be  16  sq.  ft.     Show  that 
either  dimension  varies  inversely  as  the  other. 

121.  Joint  Variation.  —  When  a  quantity  varies  directly  as  the 
product  of  two  others,  it  is  said  to  vanj  ivith  them  jointly. 

Thus,  if  2  oc  xy,  or  z  =  kxy,  then  z  varies  jointly  as  x  and  y. 

122.  Exercises. 

,'      1.    Show  that  the  area  of  a  rectangle  varies  jointly  as  its  dimensions. 
/      2.    Show  that  the  volume  of  a  right  cyHnder  varies  jointly  as  its  base  and 
altitude. 

3.  Same  as  in  2  for  a  right  circular  cone. 

4.  Show  that  the  volume  of  a  sphere  varies  jointly  as  the  radius  and  the 
area  of  a  great  circle. 


6. 
6. 
7. 

liyccx  and  x  cc  z,  show  that  y  <^  z. 
If  a;  cc  2  and  y  °^  z,  show  that  ax  + 
li  x^ccy  and  z^  oc  y,  show  that  xz  <x 

8. 

If  X  a  -  and  2/  a  -,  show  that  x  oc  2 

9.  If  X  varies  jointly  as  p  and  q,  and  y  varies  directly  as  -  ,  show  that  p^ 
varies  jointly  as  x  and  y. 

10.  According  to  Boyle's  law  of  gases,  pressure  times  volume  is  constant. 
Show  that  the  pressure  (p)  varies  inversely  as  the  volume  (v).  Show  graphi- 
cally the  relation  between  p  and  v  ii  v  =  I  cu.  ft.  when  p  =  25  lbs.  per  sq.  in. 

11.  If  w  =  uv,  show  that  w  oc  u  when  v  is  constant,  and  that  wccv  when 
u  is  constant. 

li  a :  b  =  c  :  d,  show  that 

12.  4  a  +  5  6  :  3  a  +  2  6  =  4  c  +  5  rZ  :  3  c  +  2  (f . 

13.  a-2b:  -2a+b=c-2d:  -2c+d. 

14.  7na  +  nb  :  pa  -{-  qb  =  mc  +  nd  :  pc  +  qd.  ^ 

15.  3a  +2c:a  -  c  =  3b  +2d:b  -  d. 

16.  U  -  4:  c  :  U)  -  A  d  =  2  a  +  I  c  :  2b  +  I  d. 

17.  a:a-\-c=a-\-b:  a-{-b-\-c-\-d. 


122]  RATIO,   PROPORTION,   VARIATION  93 

18.  a^  +ab  +  h"-  :  a"-  -  ab  -\- h"^  =  d^  -\- cd  +  d"- :  c"-  -  cd  +  d"^. 

19.  a  +h:c  ^-d::  yja'-  +  62  :  Vc^  +  d^. 

20.  Va^  +  62  :  Vc2  +  rf2  =  ^/aS  +  63  :  \/cM^- 

21.  VoM^  :  Vc2  +  d2  =  -s/oa  -  6^  :  Vc^  -  d^. 
If  a  :  6  =  c  :  d  and  p  :  q  =  r  :  s,  show  that 

22.  p'^a"  :  r'"6"  =  g'"c"  :  s'"^". 

23.  (o  +  6)  (p  -  r) :  (o  -  6)  (p  +  r)  =  (c  +  d)  (g  -  s) :  (c  -  d)  (5  +  s). 

Solve  for  x: 

27.  The  intensity  of  Hght  varies  inversely  as  the  square  of  the  distance 
from  the  source.  If  the  sun  is  equivalent  to  600,000  full  moons  in  brightness, 
at  how  many  times  its  present  distance  would  it  be  of  the  same  brightness  as 
the  full  moon? 

28.  The  squares  of  the  periods  of  revolution  of  the  planets  about  the  sun     \„,-- 
vary  as  the  cubes  of  their  mean  distances.     The  earth  makes  a  revolution  in 

one  year  at  a  mean  distance  of  93,000,000  miles.     Venus  makes  a  revolution 
in  225  days,  Jupiter  in  12  years.     Find  their  mean  distances  from  the  sun. 

29.  In  beams  of  the  same  width  and  thickness  the  deflection  due  to  a  cen- 
tral load  varies  jointly  as  the  load  and  the  cube  of  the  length.  If  a  beam 
10  ft.  long  is  bent  ^  inch  by  a  load  of  1000  lbs.,  how  much  will  a  load  of 
5000  lbs.  bend  a  30-ft.  beam? 

30.  Two  lights,  one  of  which  is  twice  as  strong  as  the  other,  arc  10  ft.  ^ 
apart,     Where  on  the  line  joining  them  do  they  produce  equal  illumination? 


26. 


CHAPTER   VII 

The  Trigonometric  Functions 


123.   Consider  any  number  of  right  triangles  having  a  common 
acute  angle  A,  as  ABid,  AB2C2,  and  AB3C3,  in  the  figure. 

(j^  Since  these  triangles  are  simi- 
lar, homologous  sides  are  propor- 
tional, and  therefore 


B3C3 
ACs 


=  X. 


X  (lambda)  denoting  the  common 
value   of  the   ratio   of  the   side 
opposite  Z  A  to  the  hypotenuse  in  the  several  triangles. 

Evidently,  in  every  right  triangle  having  an  acute  angle  equal  to 
A  the  ratio  of  the  side  opposite  Z  A  to  the  hypotenuse  has  the 
same  value  X;  X  depends  only  on  Z  ^,  and  not  at  all  on  the  par- 
ticular triangle  in  which  this  angle  may  be  found.     For  example,  if 


45°,  X 


V2' 


A  =  60°,  X  =  W3. 


Hence  we  see  that  X  is  a  function  of  A,  and  that  to  every  value 
of  A  corresponds  a  definite  value  of  X. 

This  function  is  called  the  sine  of  angle  A,  or 

X  =  sine  of  angle  A  =  sin  A. 

94 


124,125] 


TRIGONOMETRIC  FUNCTIONS 


95 


124.  The  ratio  of  the  side  opposite  the  angle  to  the  hypotenuse 
is  merely  one  of  six  possible  ratios  which  may  be  formed  from  the 
three  sides  of  any  right  triangle.  Hence  associated  with  every 
acute  angle  there  are  six  ratios,  or  six  abstract  numbers,  whose 
values  depend  merely  on  the  magnitude  of  the  angle.  They  are 
called  the  six  trigonometric  ratios,  or  trigonometric  functions  of 
the  angle,  and  are  named  as  follows: 

opposite  side 

hypotenuse 
adjacent  side 

hjrpotenuse 

opposite  side 
tangent  of  Z  .1  =  tan.l  =  ^^jIZi^^Hidi  * 

hypotenuse 
opposite  side 

hypotenuse 
adjacent  side 
adjacent  side 


sine  of  Z  ^  =  sin  A 


cosine  of  Z  A  =  cos  A 


cosecant  of  Z  ^  =  esc  A 


secant  of  Z  ^1  =  sec  A 


cotangent  of  Z  A  =  cot  A 


opposite  side 


If  the  sides  of  the  triangle  are  a,  b,  c,  as  in  the  figure,  then 
sm  A  =  —f      CSC  A  =  — > 


cos^l 


tan^ 


sec^  = 


cot^  = 


125.  Exercises.  With  the  aid  of  a  pro- 
tractor (see  inside  of  back  cover),  construct 
triangles  containing  the  following  angles  and, 

by  measuring  the  sides  and  dividing,  calculate  to  two  decimals 

the  six  functions  of  these  angles. 


1. 

30°. 

4. 

75°. 

7. 

85°. 

10. 

5°. 

2. 

45°. 

5. 

15°. 

8. 

80°. 

11. 

57° 

3. 

60°. 

6. 

18°. 

9. 

10°. 

12. 

38° 

Check  the  results  of  the  preceding  exercises  by  means  of  the 
following  table. 

f  Of    THf  \ 


96 


TRIGONOMETRIC   FUNCTIONS 


[126 


Angle. 

Sin. 

Cos. 

Tan. 

Cot. 

Sec. 

Csc. 

0° 
5 
10 

0.087 
0.174 

0.996 
0.985 

0.087 

0.176 

11.430 
5.671 

1.004 
1.015 

11.474 
5.759 

15 
20 
25 

0.259 
0.342 
0.423 

0.966 
0.940 
0.907 

0.268 
0.364 
0.466 

3.732 

2.748 
2.144 

1.035 
1.064 
1.103 

3.864 
2.924 
2.366 

30 
35 
40 

0.500 
0.574 
0.643 

0.866 
0.819 
0.766 

0.577 
0.700 
0.839 

1.732 
1.428 
1.192 

1.155 
1.221 
1.305 

2.000 
1.743 
1.556 

45 
50 
55 

0.707 
0.766 
0.819 

0.707 
0.643 
0.574 

1.000 
1.192 
1.428 

1.000 
0.839 
0.700 

1.414 
1.556 
1.743 

1.414 
1.305 
1.221 

60 
65 
70 

0.866 
0.906 
0.940 

0.500 
0.423 
0.342 

1.732 
2.145 

2.748 

0.577 
0.466 
0.364 

2.000 
2.366 

2.924 

1.155 
1.103 
1.064 

75 
80 
85 

0.966 
0.985 
0.996 

0.259 
0.174 
0.087 

3.732 
5.671 
11.430 

0.268 
0.176 
0.087 

3.864 
5.759 
11.474 

1.035 
1.015 
1.004 

90 

126.   Given  one  function,  to  determine  the  other  functions. — 

When  a  function  of  an  acute  angle  is  given,  the  angle  may  be 
constructed  by  writing  the  given  function  as  a  fraction,  and  con- 
structing a  right  triangle,  two  of  whose  sides  are  the  numerator 
and  denominator  of  this  fraction  respectively,  or  equal  multiples 
of  these  quantities.  Also,  since  the  third  side  of  the  triangle  can 
be  calculated  from  the  other  two,  all  the  other  functions  of  the 
angle  may  be  found  when  one  function  is  given. 


Examples. 


cot  A  =  ; 


Scaling  off  the  a-ngle  with  a  protractor,  we  have  A  =  37°.     By  taking 
from  the  table  the  angle  whose  tangent  is  .75  we  have  A  =  37°  as  before. 


1. 

A  3  /  opp.  side 
t^"^      4  1       adj.  side, 

Lay  off  AC  = 

=  4andrB  =3,  ±AC. 

Then 

AB  =  V42  +  32  =  5. 

Hence 

•  ^  3.  .4. 
sin  A  =  ^  ,   cos  A  =  -^  , 

A  5  .  5. 
csc  A  =  ;^  ;  sec  A  =  J , 

Sec  A 


3  hyp.     ' 

1       adj.  side; 


127,  128] 


TRIGONOMETRIC   FUNCTIONS 


97 


Lay  off  AC  =  1.  With  A  as  center  and  radius  =  3,  strike 
an  arc  to  cut  the  -L  drawn  to  AC  at  C.  This  determines 
the  point  B. 

The  solution  may  now  be  completed  as  in  example  1. 

Another  method  of  constructing  the  triangle  in  this 
example  is  to  calculate  CB  first,  and  then  to  proceed  as 
in  example  1. 

127.  Exercises.  Determine  the  angle  (approxi- 
mately) and  the  remaining  functions,  when 


1. 

sin  A  = 

2. 

sin  A  = 

3. 

sin  A  = 

4. 

cosA.  = 

5. 

cos  A  = 

6.  tan  A  =  2  • 

7.  tan  A  =  4. 

8.  tan  A  =  V3. 

9.  cot  A  =  1. 

10.  cot  A  =  1.5. 

11.  sec  A  =  2. 


12.  esc  A  =  2  • 

13.  cos  A  =  0.2. 

14.  CSC  A  =  2.4. 

15.  tan  A  =  10. 


16.  Show  that  the  equation  sin  A  =  2  is  impossible.^ 

17.  Show  that  the  equation  cos  A  =  1.1  is  impossible. 

18.  Show  that  the  equation  sec  A  =  \  is  impossible.  ' 

19.  Show  that  the  equation  esc  A  =  .9  is  impossible.  ' 
When  A  is  an  acute  angle  show  that, 

20.  sin  A  lies  between  0  and  l.v^ 

21.  cos  A  lies  between  0  and  1. 

22.  sec  A  and  esc  A  are  always  greater  than  1. 

23.  tan  A  and  cot  A  may  have  any  value  from  0  to  oo . 


128.  Functions  of  Complementary 
Angles.  —  Since  the  sum  of  the  two 
acute  angles  of  a  right  triangle  is  90°, 
they  are  complementary. 

By  definition,  we  have 

.     „      opp.  side      h  . 

A  o  c  sm  B  =  -^^ =  -  =  cos  A. 

hyp.         c 

Also,  cos  B  =  sin  A  ;  tan  B  =  cot  A ;  esc  J5  =  sec  ^4 ;  sec  .B  =  esc  A  ; 
and  cot  B  =  tan  .4. 

Complementary  Functions,  or  Co-functions.  —  The  co-sine  is 
called  the  complementary  function  to  the  sine  and  conversely. 
Similarly  tangent  and  co-tangent  are  mutually  complementary, 
and  secant  and  co-secant. 


98  TRIGONOMETRIC   FUNCTIONS  [129,130 

The  preceding  equations  are  now  all  contained  in  the  following 
Rule :   Any  function  of  an  acute  angle  is  equal  to  the  co-function 
of  the  complementary  angle. 

Exercise.     Verify  this  rule  when  A  =  30°,  45°,  and  60°. 

129.  Application  of  the  Trigonometric  Functions  to  the  Solution 
of  Right  Triangles.  —  When  two  parts  of  a  right  triangle  are 
known,  exclusive  of  the  right  angle,  the  triangle  may  be  constructed 
and  the  remaining  parts  determined  graphically.  By  the  aid  of 
tables  of  the  trigonometric  functions,  the  unknown  parts  may  also 
be  calculated. 

Rule :  When  two  parts  of  a  right  triangle  are  given  (the  rt.  Z 
excepted)  and  a  third  part  is  required,  write  down  that  equation 
of  (124)  which  involves  the  two  given  parts  and  the  required  part. 
Substitute  in  it  the  values  of  the  given  parts,  and  solve  for  the 
required  part. 

An  exceptional  case  arises  when  two  sides  are  given  and  the  third 
side  is  required.  In  this  case  we  may  use  the  formula  a^  -\-  b^  =  c^. 
It  will  usually  be  better  however,  unless  the  given  sides  are  repre- 
sented by  small  numbers,  to  solve  for  one  of  the  angles  first,  and 
then  to  obtain  the  third  side  from  this  angle  and  one  of  the  given 
sides. 

Example  1.  In  A  ABC,  given  A  =  40°,  C  =  90°,  and  b  =  60°.  Find  the 
other  parts  of  the  triangle. 

To  get  B,  we  have  B  =  90°  -  A  =  50°. 

To  get  a,  take  t  =  tan  A    or    a  =  b  tan  A. 
Finally,  c  is  determined  from  -  =  cos  A 

b  ^.  A 

or     c  =  r  —  osecA. 

cos  A 

From  the  table    of  (125),  tan    40°   =  0.839    and 

A  eo  c    sec  40°  =  1.305. 

Hence  a  =  60  X  0.839  =  50.340  and  c  =  60  X  1.305  =  78.300. 

As  a  check,  we  should  have 

a  ^  50.340       „„.„ 

^  =  cos5    or     78:300  =  0-643. 

130.  Exercises. 

Determine  the  unknown  parts  of  right  triangle  ABC,  C  being  90°,  from 
the  parts  given  below.  Check  results  by  graphic  solution  and  by  a  check 
formula  containing  the  unknown  parts.     (Use  the  table  of  (125).) 


TRIGONOMETRIC  FUNCTIONS 


99 


1.  A=  25°,  a  =  100. 

6.   B  =  10°,  a  =  0.15. 

.2.  A  =  70°,  b  =  150. 

7.   A  =  4'  ',  c  =  0.045 

3.  A  =  51°,  c  =  75. 

8.   B  =S5°,   c  =  1.25. 

4.  fi  =  38°,  c  =  50. 

9.   5  =  57°,  a  =  16f . 

5.  B  =  65°,  6  =  750. 

10.   A  =  20°,  6  =  ,K. 

11.  Find  the  length  of  chord  subtended  by  a  central  angle  of  100°  in  a 
circle  of  radius  50  ft.     (First  find  the  half-chord.) 

12.  Find  the  central  angle  subtended  by  a  chord  of  80  ft.  in  a  circle  of 
radius  200  ft. 

13.  Find  the  radius  of  the  circle  in  which  a  chord  of  100  ft.  subtends  an 
angle  of  70°. 

14.  Find  the  length  of  side  of  a  regular  pentagon  inscribed  in  a  circle  of 
radius  500  ft. 

15.  Find  the  length  of  side  of  a  regular  decagon  circumscribed  about  a 
circle  of  radius  100  ft. 

16.  From  a  point  in  the  same  horizontal  plane  as  the  foot  of  a  flag-pole, 
and  300  ft.  from  it,  the  angle  of  elevation  of  the  top  is  22°.  How  high  is  the 
pole? 

17.  A  vertical  pole  75  ft.  high  casts  a  shadow  60  ft.  long  on  level  ground. 
Find  the  altitude  of  the  sun. 


131.   Angles  of  any  Magnitude,  Positive  or  Negative.  —  Con- 
sider Z  XOP  (figure)  as  generated  by  a  moving  line  which  rotates 
about  0  from  the  position  OX  to  the  posi- 
tion OP.  ^ 

Divide  the  plane  into  four  quadrants  (I, 
II,  III,  and  IV  in  the  figure '» below)  by 
means  of  two  rectangular  axes  OX  and  OY. 

In  the  figure,  the  arrows  on  the  axes 
indicate  the  positive  directions,  and 
quadrant  I  is  that  included  between 
the  positive  parts  of  the  axes.  Let 
a  moving  line  start  from  the  posi- 
tion OX  and  rotate  into  the  positions 
OF^,  OP 2,  OP3,  and  OP4  successive- 
ly, thus  generating  the  angles  XOP\ , 
XOP2,  XOP:i,  and  XOP4  respec- 
tively. 

OX  is  called  the  initial  line,  and 
OPx  the  terminal  line  of  the  angle  XOPi,  and  similarly  for  any 
other  angle. 


100 


TRIGONOMETRIC   FUNCTIONS 


[132 


An  angle  is  positive  when  the  generating  hne  rotates  counter- 
clockwise (in  the  direction  of  the  curved  arrow  in  the  figure) 
negative  when  the  generating  line  moves  clockwise. 

The  quadrant  of  an  angle  is  that  quadrant  in  which  its  terminal 
line  lies.     The  angle  is  said  to  lie  in  this  quadrant. 

The  initial  line  OX,  and  any  terminal  line,  as  OPo,  may  always 
be  considered  to  form  two  angles  numerically  <  360°,  as  +120° 
and  —240°  in  the  figure. 

When  the  moving  line  rotates  from  OX  through  more  than  one 
complete  revolution,  an  angle  greater  than  360°  is  generated. 
Thus  a  rotation  in  the  positive  direction  (positive  rotation)  through 
H  revolutions  generates  an  angle  of  480°,  lying  in  the  second 
quadrant;  a  negative  rotation  through  2^  revolutions  generates 
an  angle  of  —780°,  lying  in  the  fourth  quadrant. 

132.  The  Trigonometric  Func- 
tions of  any  Angle. — Let  XOP  be 
any  angle,  and  P  any  point  in  its 
terminal  line.  (The  four  possible 
cases  are  here  shown  in  the  figure, 
according  to  the  quadrant  of  the 
angle.)  Let  OM  be  the  abscissa 
of  P,  MP  (not  PM)  the  ordinate 
of  P,  and  OP  the  distance  of  P. 
The  signs  of  these  quantities  are 
taken  according  to  the  usual 
convention  and  are  shown  in 
the  figure.     We  now  define  the 


functions  of   angle  XOP,  in 

whatever  quadra 

nt  it  may  1 

.     ,.^  „      ordmate  (of  P) 

sm  xor  =  -jr- ;  ,     ' ; 

distance  (of  I*) 

CSC  XOP 

distance 
ordinate ' 

,.^  ^.      abscissa 

cos  xor  =  -rr- ; 

distance 

sec  XOP 

distance 
abscissa ' 

abscissa 

cot  XOP 

abscissa 
ordinate 

When    Z  XOP  <  90°,    these    definitions    agree    with    those 
of (124). 


133,134] 


TRIGONOMETRIC   FUNCTIONS 


101 


According  to  the  above  definitions  we  have  the  following 
Table  of  Signs  of  the  Trigonometric  ''unctions 


Quadr. 

sin. 

COS. 

tan. 

cot. 

sec. 

CSC. 

I 

+ 

+ 

+ 

+ 

+ 

+ 

II 

+ 

— 

— 

— 

— 

+ 

III 

— 

— 

+ 

+ 

— 

IV 

- 

+ 

+ 

- 

Let  the  student  verify  carefully  the  signs  in  this  table.  He 
should  be  prepared  to  state  instantly  the  sign  of  any  function  in 
any  quadrant. 

Observe  that  in  the  first  quadrant  all  the  functions  are  positive; 
in  the  other  quadrants  a  function  and  its  reciprocal  are  positive, 
the  remaining  four  negative. 

133.  Approximate  Values  of  the  Functions  of  any  Angle.  — 
If  in  the  last  figure  the  distances  OP  had  been  taken  all  of  the  same 
length,  all  the  points  P  would  lie 
on  the  circumference  of  a  circle 
with  center  at  0. 

Let  us  draw  a  circle  with  0  as 
center  and  unit  radius  (figure; 
1  =  10  small  divisions).  Then  for 
any  angle  XOP  we  have 

Mpy 


sin  XOP  =  MP 


cos  XOP  =  0M 


1   / 


i-r) 


Hence  approximate  values  of  the  sines  and  cosines  of  all  angles 
may  be  read  off  directly  from  the  figure.     The  other  functions 

MP 
may  be  obtained  by  division,  since  tan  XOP  =  ^^rj,  etc.     They 

may  also  be  constructed  graphically  by  a  method  explained  in  the 
next  article. 

The  lines  OM  and  MP,  whose  lengths  represent  the  sine  and 
cosine  of  Z  XOP,  are  'commonly  referred  to  as  the  line  values  of 
these  functions. 

134.  Line  Values  of  the  other  Trigonometric  Functions.  — 
Construct  a  circle  as  in  the  figure  below  and  draw  the  tangents 


102 


TRIGONOMETRIC   FUNCTIONS 


[134 


at  S  and  *S'.     Let  XOP  be  an  angle  in  the  first  quadrant.     Pro- 
duce OP  to  meet  t^e  tangent  at  S  in  T.     Then  by  similar  triangles, 

MP      ST 


Ts 

Y 

/ 

Tz 

v-"^^ 

/7y 

T 

£. 

I 

\\ 

f 

\] 

T, 

/^V^^^^^ 

> 

/ 
\ 

Ts 

tan  XOP 


OM      OS 


ST. 


In  the  same  way, 

tanZOPi  =STi; 
tan  XOP2  =  ST.. 
Hence  when  an  angle  is  in  the 
first    quadrant,    its    tangent    is 
measured  by  the  segment  of  the 
tangent  line  from  S  to  the  ter- 
minal line  produced;  the  radius 
of  the  circle  is  the  unit  of  length. 
By    taking    into   account  the 


algebraic  sign  of  the  tangent,  we  find  that 

tan  XOP3  =  -  S'T^;  tan  XOP4  =  -  ^'^4;  tan  XOP 5 


ST, 


Here  ST4  and  ST5  are  themselves  negative  lines. 

Hence  the  numerical  value  of  the  tangent  of  any  angle  equals 
the  segment  of  the  vertical  tangent  cut  off  by  the  terminal  line 
produced,  this  segment  being  measured  in  terms  of  the  radius  as 
unity.  This  value  should  be  given  the  proper  sign  according  to 
the  quadrant  of  the  angle. 
^     W,e  have  further. 


sec  XOP  = 


OP^ 
OM 


OT 

^^  =  OT 

OS      ^ 


By  examining  the  other  angles  shown  in  the  figure  we  see  that 
the  numerical  value  of  the  secant  of  any  angle  equals  the  segment 
of  the  terminal  line  produced  from  the  origin  to  the  vertical  tan- 
gent. The  proper  sign  may  be  determined  according  to  the 
quadrant  of  the  angle. 

To  obtain  graphic  constructions  of  the  cotangent  and  cosecant, 
we  draw  the  tangents  at  R  and  R'  (figure  below).     Then 


cot  XOP 


CSC  XOP 


OM^ 
MP 
OP 
MP 


m-^^' 


OT 
OR 


=  OT. 


135] 


TRIGONOMETRIC   FUNCTIONS 


103 


By  examining  the  other  angles  in  the  figure  we  see  that,  (a)  the 
cotangent  of  any  angle  is  numerically  equal  to  the  length  of  the 
segment  of  the  horizontal  tan- 
gent cut  off  by  the  terminal 
line  of  the  angle  produced; 
(b)  the  cosecant  is  numerically 
equal  to  the  segment  of  the 
terminal  line  produced  from 
the  origin  to  the  horizontal 
tangent. 

In  either  case  the  sign  is  to 
be  determined  according  to 
the  quadrant  of  the  angle. 
N  135.  Variation  of  the  Trigo- 
nometric Functions.  —  In  the  figure  of  (133)  suppose  the  point 
P  to  describe  the  circumference  of  the  circle  in  such  a  way  that 
the  angle  XOP  shall  vary  continuously  from  0°  to  360°.  Let  us 
trace  the  changes  in  the  value  of  sin  XOP  =  MP.  .  In  the  first 
quadrant  MP,  and  hence  sin  XOP,  varies  from  0  to  +1,  in  the 
second  from  -|-1  to  0,  in  the  third  from  0  to  -  1  and  in  the 
fourth  from  —  1  to  0. 

Similarly  cos  A'OP  varies  in  the  four  quadrants  successively 
from  +1  to  0,  0  to  -1,  -1  to  0,  and  Oto  +1. 

Consider  next  tan  XOP=^'    When  XOP  =  0°,  MP  =  0  and 

CM  =  1 ;  hence  tan  0°  =  0. 

Now  as  XOP  increases  from  0°  toward  90°,  MP  steadily  increases 
toward  1,  while  OM  steadily  diminishes  toward  0.  Hence  tan 
XOP  increases  from  0  without  limit,  so  that  we  write  tan  90°=  co, 
and  say  that  the  tangent  varies  from  0  to  ^  as  XOP  varies  from 
0°  to  90°. 

Since  the  three  remaining  functions  are  reciprocals  of  the  three 
already    considered,    their   variations   are   easily   traced.     Thus, 


CSC  XOP 


Hence  esc  XOP  varies  from  «  to  1  in  the 


sin  XOP 

first  quadrant,  and  from  1  to  <»  in  the  second.  Now  as  XOP 
passes  through  180°,  esc  XOP  changes  suddenly  from  a  large  posi- 
tive value  when  the  angle  is  a  little  less  than  180°  to  a  large 
negative  value  when  the  angle  is  a  little  more  than  180°. 


104 


TRIGONOMETRIC  FUNCTIONS 


[135 


This  abrupt  ci^iange  in  the  cosecant  when  the  angle  passes 
through  180°  is  ex) pressed  by  saying  that  the  cosecant  has  a  dis- 
continuity at  180°;  5'^ec  180°  may  be  either  +oc  or  -oo,  according 
to  the  side  from  which  XOP  approaches  180°. 

In  the  third  quadrant  esc  XOP  is  negative  and  varies  from 
—  00  to  —1;  in  the  fourth  quadrant  from  —1  to  —  oo.  There  is 
another  discontinuity  at  360°  or  0°, 

The  variations  of  the  six  functions  are  shown  in  the  following 
table.        ,  . 


Quadr. 

sin. 

CSC. 

COS. 

sec. 

tan. 

cot. 

II 
III 
IV 

Oto,  +  1 
+1  toO 

Oto  -1 
-1  toO 

+  C0  to  +1 

+  1      to    +00 

-co  to  -1 
-1    to  -oo 

+  1  too 
oto  -1 

-1  toO 
oto  +1 

+  1      to   +00 

-00  to  -1 
-1     to  -00 
+  00  tol 

0  to  +00 
-ootoO 

0  to  +00 
-ootoO 

+00  toO 
0  to-x 
00  toO 
Oto-oo 

The  range  of  values  covered  by  each  of  the  six  functions  is  indi- 
cated in  the  diagram  below. 


sec  X  and  esc  x 


sin  X  and  cos  x 


sec  X  and  esc  x 


tan  X  and  cot  x 


Exercises. 


1.  Trace  carefully  the  variations  given  in  the  above  table. 

2.  Show  that  the  following  functions  have  discontinuities  at  the  values 
stated:  the  tangent,  at  90°  and  270°;  the  cotangent,  at  0°  and  180°;  the  secant, 
at  90°  and  270°;  the  cosecant  at  0°  and  180°. 

3.  Discuss  the  "equations,"  tan  90°  =  +oo ;  tan  90°  =  -oo.  Same  for 
csc0°  =  +00 ;  cscO°  =  -oo. 

4.  Draw  a  circle  as  in  the  figure  of  (133),  adding  also  the  vertical  and  hori- 
zontal tangents.  Divide  the  circumference  into  36  equal  parts,  and  obtain 
from  the  diagram  a  two-place  table  of  the  six  functions  for  every  tenth  degree 
from  0°  to  360°. 

,5.  By  use  of  the  results  of  exercise  4,  trace  the  graph  of  the  equation  y=sm  x. 
[Take  angle  x  on  a  horizontal  scale,  making  one  small  square  =  10°.  On 
the  vertical  scale  choose  any  convenient  length  as  1  (=  sin  90°),  say  10  small 
squares.  At  successive  points  x  on  the  horizontal  axis  erect  perpendiculars 
equal  to  sin  x,  upward  or  downward  according  to  the  sign.  See  note  at  end 
of  (143)]. 


136] 


TRIGONOMETRIC  FUNCTIONS 


105 


Graphs  of  the  Trigonometric  Functions 


Sine  Curve        *i 
(full  line) 

Cosine  Curve       o 
(broken  line) 


Tangent  Curve    -^i 
(full  line) 

Cotangent  o 

Curve 
(broken  line) 


Secant  Curve      h 
(full  line) 
Cosecant  Curve     o 
(broken  line) 


I 


4^0"  [S^O" 


106  TRIGONOMETRIC   FUNCTIONS  [136,137 

6.  Trace  the  graph  of  ?/  =  cos  x.     (On  same  diagram  as  ?/  =  sin  x.) 

7.  Trace  the  graphs  of  y  =  tan  x  and  y  =  cot  x. 

8.  Trace  the  graphs  of  y  =  sec  x  and  7/  =  esc  x. 

'^ISG.  Periodicity  of  the  Trigonometric  Functions.  —  Since  the 
position  of  the  terminal  Hne  of  an  angle  x  is  unchanged  when  the 
angle  is  increased  or  diminished  by  integral  multiples  of  360°,  any 
function  of  x  equals  the  same  function  of  x  ±  7i.dQ0°,  n  being  an 
integer.     That  is, 

fix)  =/(x±n.360°), 

where  /  stands  for  any  one  of  the  trigonometric  functions. 

Hence  the  trigonometric  functions  are  periodic,  with  a  period 
of  360°.     (See  graphs  on  p.  105.) 

137.  Relations  between  the  Functions  of  an  Angle.  —  From  the 
general  definitions  of  the  functions  given  in  (133)  we  have,  putting 
Z  XOP  =  X, 

1  1^1 

sin  -JO  = ;     cos  x  ^  ;     tan  oc  =  — —- . 

CSC  X  sec  ic  cot  X 

ordinate 

ordinate      distance      sin  .r .       ^  cos  x 

tan  X  =  — j — ; =  — r — r— -  =  ,   cot  x  =  -: • 

abscissa        abscissa      cosic  sin  a? 


distance 


Whatever  be  the  quadrant  of  angle  XOP  =  x  [figure  of  (132)], 
we  have 

(ordinate)^  +  (abscissa)'^  ==  (distance)'. 

Dividing  this  equation  through  in  turr^  by  (distance)-,  (abscissa)-, 
and  (ordinate)-,  and  expressing  the  resulting  ratios  as  functions 
we  have 

sin^ic+  cos^cc  =  1,  ■ 
1  +  tan'  X  =  sec'  x, 

1  +  COt^  X  —  CSC'  X. 

All  the  above  relations  between  the  functions  of  an  angle  x  are 
true  for  all  values  of  x.  They  form  a  first  set  of  working  formulas, 
and  should  be  thorouglily  committed  to  memory.  They  are 
collected  below,  as 


138]  TRIGONOMETRIC   FUNCTIONS  107 

Formulas,  Group  A. 

,,^    .  1  ...  .  since        (6)  sin-x  +  cos-iT  =  1. 

(l)sinx  = (4)tanx  = ^ 

CSC  a;  cos  J)         /7N  .    ,   *     •• 

(7)  1  +  tan-.r  =  sec'iT. 
/ox  1  /r\       X  COS.r 

(2)  COSX  = (5)  cot  .*•  =  -: /o\   i  _L  ^^4-'i  -^^^-J 

secx  ^  sinx        (8)  1 +  cot'x  =  csc-u^. 

(3)  tan.r=  — ^ — 

cotoj 

We  shall  apply  these  formulas  in  two  examples. 
Example  1.    Prove  that  tan  x  +  cot  x  =  sec  x  esc  x. 

sin  X   .    cos  a;       sin2  x  +  cos^  x 

tan  X  +  cot  X  = 1 — -. —  =  — -. — 

cos  X       sm  X  sm  x  cos  x 


sm  X  cos  X 

Example  2.     Prove  that 
esc  X 

;r j r-   =  cos  X. 

tan  X  +  cot  X 


CSC  X  sec  X. 


tan  X  +  cot  X        sin  .c  ,   cos  x       sin2  x  +  cos^  x 


cos  X      sm  X  sm  x  cos  x 


sm  X  cos  X 


CSC  X  sm  X  cos  x  =  cos  x. 


In  both  examples  all  the  steps  taken  are  true  for  all  values  of  x,  since 
this  is  true  of  all  the  formulas  of  group  A.  Hence  the  given  equations  are 
true  for  all  values  of  x,  and  they  are  therefore  called  trigonometric  idcntilies. 

The  equation  sin^  x  —  cos^  x  =  1  is  not  true  for  all  values  of  x,  but  holds 
only  for  certain  special  values;  it  is  not  an  identity. 


138.   Exercises.     Prove  the  following  identities: 

1.  tan  X  cos  x  =  sin  x.  .        ,  esc  x 

4.   cot  X  = 

J  sec  X 

2.  -— =  sin  X.  e     /  •  0      1        •>    NO       I 

cot  X  sec  X  5.    (sm^  X  +  cos- .r)-  =  1. 

«     .  sec  X  _  cos  9  ,  „  - 

3.  tanx  = 6.    -^— rr — ^  =  cot2». 

esc  X  sm  6  tan  0 

7.  (esc  e  —  cot  e)  (esc  0  +  cot »)  =  1. 

8.  (sec  X  —  tan  x)  (sec  x  +  tan  x)  =  1. 

9.  (sin  9  -f  cos  5)2  =  1  +  2  sin  d  cos  0. 

10.  sin2  a  +  cos2  a  =  sec2  a  —  tan2  a. 

11.  (sina  —  cosa)2  =  1  —  2sinQ:cosa. 


108  TRIGONOMETRIC   FUNCTIONS  [139 

12.  sin4  X  —  cos4  x  =  sin2  x  —  cos2  x. 

13.  (1  -  sin2  x)  csc2  x  =  cot2  x. 

..-'^4.     C0t2  e  -  C0S2  e  =  C0t2  0  C0S2  6. 

15.  tan  0  +  cot  0  =  sec  d  esc  9. 

16.  tan  ^  sin  ^  +  cos  </>  =  sec  9!>. 

17.  sin2  0  sec2^  =  sec2  <j)  -  \.  20.   (1  -  sin2  /3)  (1  +  tan2  /3)  =  1. 

18.  ^^^''^      =  i±.C08^^  21.   tan"  x  -  sec"  x  =  1  -  2  sec2  x. 

'  1  —  cos  ^  sin  <yi 

1    ,x     o^        -o^                          _„    cosx  +  sinx       1+tanx 
.-    l+tan2/3      sm2/3  22.   ■ — -. —  = -— !— 

19.  T— ; -r-^  =  — r-^  •  COS  X  —  sin  X      1  —  tan  x 

l+C0t2;S  COs2/3 

23.    (tan  x  —  1)  (cot  x  —  1)  =  2  —  sec  x  csc  x. 


24. 


COS( 


25.  sec 0 sin' 5  =  (1  +cos^)  (tan 5  —  sine). 

26.  tan2  a  +  cot2  a  +  2  =  sec2  a  cs(fl  a. 

27.  sin3  e  +  cos3  6  =  (sin  0  +  cos  0)  (1  —  sin  ^  cos  0). 

28.  (sin2  e  -  c'os2  (9)2  =  1  -  4  cos2  ^  +  4  cos*  (?. 

29.  sin6  e  +  cos^  e  =  sin*  0  +  cos*  e  —  sin2  ^  cos2  e. 

30.  (sin  X  —  cos  x)  (sec  x  —  csc  x)  =  sec  x  csc  x  —  2. 
„^  tan  X  —  cot  X  _      2     _ 

tanj;  +  cotx      csc2x 

32.  (a  cos  X  -  6  sin  x)2  +  (a  sin  x  +  h  cos  0)2  =  a2  +  }fi. 

33.  cos2  <i6  4-  (sin  0  cos  0)2  +  (sin  0  sin  0)2  =  1. 

34.  tan  a  +  tan  /S  =  tan  a  tan  /3  (cot  a  +  cot  /3). 

139.  Functions  of  any  Angle  in  Terms  of  Functions  of  an  Acute 
Angle.  —  It  is  possible  to  express  in  a  simple  manner  any  function 
of  any  angle  in  terms  of  a  function  of  an  acute  angle.  Therefore 
a  table  of  values  of  the  functions  of  angles  from  0°  to  90°  will  serve 
for  all  angles.  In  fact,  in  view  of  (128),  a  table  of  functions  from 
0°  to  45°  would  be  sufficient,  though  not  convenient. 

1.  Any  angle,  positive  or  negative,  can  be  brought  into  the  first 
quadrant  by  adding  to  it,  or  subtracting  from  it,  an  integral  mul- 
tiple of  90°. 

Thus:        .    760°  -  8  X  90°  =  40°;     -  470°  +  6  X  90°  =  70°. 

2.  When  an  angle  is  changed  by  an  integral  multiple  of  90°, 
say  n  X  90°,  the  new  terminal  line  lies  in  the  same  line  as  the  origi- 
nal terminal  line  when  n  is  even;  at  right  angles  to  it  when  n  is  odd. 

3.  Twoangles  which  differ  by  an  eyenrnw^^ipZeo/ 90°  will  be  called 
symmetrical  with  respect  to  the  initial  line,  or  simply  symmetrical; 
two  angles  which  differ  by  an  odd  multiple  of  90°,  skew-symmetrical. 


139] 


TRIGONOMETRIC   FUNCTIONS 


109 


4.  When  two  angles  are  symmetrical,  any  function  of  the  one  is 
numerically  equal  to  the  same  function  of  the  other. 

From  figure  (a),  sin  x  =  -  sin  x'  =  sin  x",  etc.,  for  the  other 
functions. 


\ngles  x'  and  x"  are  symmetrical  with 
respect  to  angle  x 


Angles  x'  and  x"  are  skcu'-sym  metrical 
with  respect  to  x 


When  two  angles  are  skew-symmetrical,  any  function  of  the  one 
is  numerically  equal  to  the  co-function  of  the  other. 

From  figure  (b),  sina;'=  -  cos  a:'  =  cos  a;",  etc.,  for  the  other 
functions. 

Exercise  1.  From  figures  (a)  and  (b),  write  down  all  the  functions  of  x 
in  terms  of  functions  of  x'  and  of  x". 

Exercise, 2.  Draw  figures  corresponding  to  figures  (a)  and  (6),  when  x  lies 
in  each  of  the  other  quadrants.     Then  proceed  as  in  exercise  1. 

5.   Rule:  Any  function  of  any  angle  x  is  numerically  equal  to 

^,    { same  function    .      .  .        ,-..,,,  {even 

the  {         /    ,.         of  X  increased  or  diminished  by  any  <     , ,  mul- 
(    co-function  t>       u  ^  ^^^ 

tiple  of  90°. 

As  an  equation, 


^  ±  f(x  ±  n-90°),        neven; 
)  ±  co-f(x  ±  »^.90°),  nodd. 


The  sign  of  the  result  must  be  determined  by  noting  the  quadrants 
of  X  and  x  ±  n  '  90°. 


no 


TRIGONOMETRIC   FUNCTIONS 


[140,  141 


When  the  new  angle,  a;  ±  n  •  90°,  lies  in  the  first  quadrant,  give  to 
the  result  the  sign  of  the  given  function  of  x,  f  (x). 
Examples. 

1.  sin  680°  =  sin  (50°  +  7  X  90°)  =  -  cos  50°. 

Here  we  diminish  the  given  angle  by  an  odd  multiple  of  90°,  hence  change 
to  the  co-function.     Also  sin  680°  is  negative,  hence  we  use  the  minus  sign. 

2.  tan  (-  870°)  =  tan  (30°  -  10  X  90°)  =  +  tan  30°. 

3.  sec  420°  =  sec  (60°  +  4  X  90°)  =  +  sec  60°. 

140.  Relations  between  the  Functions  of  +x  and  —x.  —  The 

figure  shows  two  cases,  x  in  the  first  quadrant  and  x  in  the  second 
quadrant.     In  either  case, 

sin  a;  =  —  sin  {—  x); 
esc  a;  =  —  CSC  (—  x); 
cos  a;  =  cos  ( —  a;) ; 
sec  X  =  sec  {—  x); 
tan  X  =  —  tan  (—  x); 
cot  a;  =  —  cot  (—  a;). 

Exercise.  Show  that  these  equations 
are  true  when  x  lies  in  the  third  quadrant 
or  fourth  quadrant. 


Rule:  The  cosine  or  secant  of  any  angle  is  equal  to  the  cosine 
or  secant  respectively  of  the  negative  angle;  the  remaining  four  func- 
tions of  the  angle  are  equal  to  the  negative  of  the  corresponding  functions 
of  the  negative  angle.     Or, 


f{x)  =  /(—  x)  when  f  stands  for  cos.  or  sec. 
f  {pc>)  =  —  /  ( —  ic)  when  f  stands  for  sin.,  esc. 


tan.,  or  cot. 


-^    141.   Exercises.     Express   all    the   functions  of   the   following 
angles  in  terms  of  functions  of  acute  angles: 

1.  130°.                 5.        359°.              9.    -  321°.  13.    -  1060°. 

2.  165°.                 6.    -    25°.            10.         742°.  14.    -    401°. 

3.  230°.                 7.    -  12.5°.           11.    -  665°.  15.           525°. 

4.  340°.                 8.    -  250°.           12.       1100°.  16.    -     101°. 

Express  all  the  functions  of  the  following  angles  in  terms  of  functions  of 
angles  between  0°  and  45°. 

17.  75°.           19.    110°.               21.    -335°.  23.        790°. 

18.  -  80°.           20.   255°.               22.        600^  24.    -  510°. 


142,  143) 


TRIGONOMETRIC   FUNCTIONS 


111 


the  values 

of  the  functions  of: 

25.    120°. 

29.    -    30°. 

33. 

-  240°. 

26.    135°. 

30.    -    45°. 

34. 

315°. 

27.    150°. 

31.    -    60°. 

35. 

600°. 

28.   300°. 

32.    -  120°. 

36. 

-  510°. 

Find  the 

i.  120°.  29.  -    30°. 

1.  135°.  30.  -    45°. 

'.  150°.  31.  -    60°. 

1.  300°.  32.  -  120°. 

142.  Versed  Sine  and  Coversed  Sine.  —  The  expressions 
1  —  cos  a:  and  1  —  sin  a:  occur  often  enough  in  the  applications 
of  trigonometry  to  warrant  the  use  of  special  symbols  for  them. 
These  are 

1  —  cos  X  =  versed  sine  of  oc  =  vers  x; 

1  —  sin  x  =  coversed  sine  of  «  =  covers  oc. 

Their  line  values  are  (figure),  versa:  =  MN,  covers  a;  =  HK, 
X  being  in  the  first  quadrant. 

Exercises.     Find  the  values  of  the 
versed  sine  and  coversed  sine  of: 


1. 

30°. 

7. 

150°. 

2. 

45°. 

8. 

-    30°. 

3. 

60°. 

9. 

-  120°. 

4. 

90°. 

10. 

-  225°. 

6. 

120°. 

11. 

-  300°. 

6. 

135°. 

12. 

-  315°. 

143.   Radian  Measure.  —  The 

degree  is  an  artificial  unit  for 
the  measurement  of  angles.  In 
France,  where  at  the  time  of  the 
Revolution  an  attempt  was  made  to  put  all  measurements  on  "the 
basis  of  the  decimal  scale,  the  quadrant  of  the  circle  was  divided 
into  100  equal  parts  and  the  apgle  subtended  at  the  center  by  one 
part  called  a  grade.  Each  grade  was  then  subdivided  into  100  equal 
parts  called  minutes,  and  each  minute  into  100  seco7ids.  The 
degree  and  the  grade  are  thus  two  arbitrary  units  for  the  measure- 
ment of  angles,  and  any  number  of  such  units  might  be  chosen. 

There  is  one  unit  which  is  naturally  related  to  the  circle,  and 
which  is  as  commonly  used  in  theory  as  the  degree  in  practice. 
It  is  the  central  angle  subtended  by  an  arc  equal  in  length  to  the 
radius  of  the  circle,  and  is  called  a  radian  (figure,  p.  112). 

Since   the    circumference    contains   the   radius  2t  times,  the 
entire  central  angle  of  360°  contains  2  tt  radians,  i.e., 
2  TT  radians  =  360°. 


112 


TRIGONOMETRIC   FUNCTIONS 


[144 


Hence, 

T  radians  =  180 


radians  =  90°; 

radians  =  45°;    and  so  on. 


In  dealing  with  angles  measured 
in  radians  it  is  customary  to  omit 
specifying  the  unit  used;  it  is  under- 
stood that  when  no  unit  is  indicated 
the  radian  is  implied.     Thus,  2t  =  360°,  w  =  180°, 

-  =  60°,  2|  =  2|  radians,    and  so  on. 
o 

Note.  To  get  the  true  form  of  the  graphs  of  the  equations  y  =  sinx, 
y  =  cosx,  etc.,  take  x  in  radians  on  thex-axis,  thus:  x  =  0.1,  0.2,  0.3,  .  .  .  ,  1, 
.  .  .  and  find  the  corresponding  values  of  y;  use  the  same  unit  of  length  for 
both  X  and  y.    See  graphs  on  p.  105. 

144.  Radians  into  degrees,  and  conversely. 
Since  2  t  (radians)  =  360°, 

360°      180°  _      180° 
2x   ~    IT    "3.1416- 

also,  1  degree  =  ^  (radians)  =  -—  (radians) 


therefore,     1  radian 


57°.29+; 


(radians)  =  .017  + (radians). 


57.29  + 
Rule:   To  convert  radians  into  degrees,  multiply  the  number  of 


radians  by 


180 


or  57.29  +  . 


To  convert  degrees  into  radians,  multiply  the  number  of 
or  .017  +  . 


'  180  ^^57.29- 

By  taking  a  sufficiently  accurate  value  of  tt,  we  find, 

1  radian  =  57°.2957795  =  3437'.74677  =  206264".8. 
1°  =  .0174533  radians. 

1'  =  .0002909  radians  (point,  3  ciphers,  3,  approx.). 
1"=  .0000048  radians  (point,  5  ciphers,  5,  approx.). 


145,146]  TRIGONOMETRIC   FUNCTIONS  113 

The  measure  of  an  angle  in  radians  is  often  called  the  circular 
measure  of  the  angle.  y 

145."  Exercises.  Reduce  to  degrees,  minutes  and  seconds  the 
angles  whose  circular  measures  are: 

^     IT      Sir      5ir      5ir      7  ir 

8'  T'  T'  T'  T' 
1     1     7 


2.    1     2 

i,   -'  2     3     4 


11  ^1     ^^1      2  7r+3 

1.^    -  +  1,   2  +  3'       -6— 

6        TT        IT    —   6 

ir2  TT  +  1 


7r2  +  ll-7r7r-l 

Reduce  the  following  angles  to  circular  measure: 

6.  30°,    120°,    150°,   225°,    -60°. 

7.  375°,  -  22i°,  187°.5,  106°,  93°  45'. 

8.  85°,  191°  15',  5°  37'  30",  90°  37'  30". 

9.  10',  10",  0".l,  12°  5'  4",  21° 36'  8".l. 

10.  If  the  radius  of  the  earth  be  taken  as  3960  miles,  find  the  number  of     l^ 
feet  in  an  arc  of  1"  of  the  meridian. 

11.  How  many  radians  in  a  central  angle  subtended  by  an  arc  75   ft. 
long,  the  radius  of  the  circle  being  50  ft.? 

12.  How  many  radians  in  the  central  angle  subtended  by  the  side  of  a 
regular  inscribed  decagon? 

13.  A  wheel  makes  1000  revolutions  a  minute.     Find  its  angular  velocity 
in  radians  per  second. 

14.  If  the  angular  velocity  of  a  wheel  is  10  tt  radians  per  second,  how  many 
revolutions  per  minute  does  it  make? 

146.   Angles  Coiresponding  to  a  Given  Function. — Let  n  denote 
an  integer   positive  or  negative,  or  zero;  then  2  n  is  always  even, 
and  2  n  +  1  odd;  hence  the  angle 
2  7nr    has    the    terminal    line   OX 
(figure)  coincident  with  the  initial 
line,  and  angle  (2  ?i  +  1)  tt  has  the     x'    cn-^-mr 
terminal  line  OX'. 

Suppose  now  we  wish  to  write 
down    all    angles    x    such     that 

sin  a;  =  |.  Corresponding  to  a  given  function,  there  are  always 
(except  when  the  angle  is  a  multiple  of  90°)  two  angles  less  than 
360°;  in  this  case  they  are 

30°    and    tt  -  30°. 


114 


TRIGONOMETRIC   FUNCTIONS 


[146 


All  angles  with  the  same  terminal  line  as  either  one  of  these  will 
have  the  same  functions;  all  such  angles  are 

2  riT  +  30°  and  2  nx  +  (tt  -  30°) 
=  (2  n  +  1)  TT  -  30°. 

Hence  all  solutions  of  the  equation 


"*■  sm  a;  =  ^  are  given  by 

a;  =  2  nvr  +  30°  or  (2  n  +  1>  tt  -  30° 

In  general,  if  d  denote  the  smallest  positive  angle  whose  sine 
is  a,  then  all  solutions  of  the  equation 

(1)  sin  a;  =  a  are  x  =  2  tit  -h  0   and    (2  n  +  1)  tt  —  ^. 

Hence   also,   if   0   denote   the   smallest   positive   angle   whose 
cosecant  is  a,  the  solutions  of  the  equation 

(2)  CSC  x  =  a     are    x  =  2  tit  -\-  6    and     (2  n  +  1)  tt  —  ^. 
Consider  next  the  equation 

COS  X  =  h. 
The  two  simplest  solutions  are 

a:  =+60°     and     a;  =-60°. 
All  possible  solutions  are  given  by 

X  =  2  WTT  +  60°   and   x  =  2  nir  -  60°, 
or  X  =  27nr  ±  60°. 

In  general,  if  6  be  the  smallest  positive 
angle  whose  cosine  is  a,  all  solutions  of  the 
equation 

(3)  cos  a;  =  a     are     x  =  2  mr  ±  0. 

Hence  also,  if  0  be  the  smallest  angle  whose  secant  is  a,  all 
solutions  of  the  equation 

(4)  sec  X  =  a     are     x  =  2  ht  ±  6. 

Finally  consider  the  equation 

tan  a;  =  1. 
The  two  simplest  solutions  are 

X  =  45°     and     a;  =  tt  +  45°, 


147] 


TRIGONOMETRIC   FUNCTIONS 


115 


and  all  possible  solutions  are 

x  =  2mr-\-45°     and     z  =  2  rnr  +  (tt  +  45°), 

the  second  set  being  the  same  as  a:  =  (2  n  +  1)  w  +  45°. 

Both  sets  are  contained  in  the  single 
equation 

X  =  TIT  -{-  45°, 

the  first  set  being  obtained  when  n  is 
even,  the  second  set  when  n  is  odd. 

In  general,  if  6  be  the  smallest  posi- 
tive angle  whose  tangent  is  a,  all 
solutions  of  the  equation 

(5)  tan  X  =  a     are     x  =  mr  -{■  0. 
Hence  also,  if  0  be  the  smallest  positive  angle  whose  cotangent 

is  a,  all  solutions  of  the  equation 

(6)  cot  X  =  a  ,  are     x  =  nr -'r  0. 

Summary  of  equations  (1)  to  (6). 

Let  0  denote  the  smallest  positive  angle  having  a  given  function 
equal  to  a  given  number  a. 


11. 


III. 


^  sin  X  =  a 
I  CSC  Jc  =  a 

(  cos  a?  =  a 
I  sec  X  ^  a 
( tan  X  =  a 
)  cot  J^  =  a 


Then  all  solutions  of  the  equation 
are     £c  =  2  jtir  +  6    and    (3n.  +  l)TT  — 6; 

are     x  =  2  uir  ±  6 ; 

are     x.  =  »</t7  +  0. 


The  angle  0  is  usually  called  the  principal  value  of  x. 

The  solutions  of  these  equations  may  also  be  written  by  the 
following  simple  rule. 

Rule:  Corresponding  to  a  given  value  of  a  function,  there  are 
in  general  two  and  only  two  positive  angles  less  than  360°.  If 
these  be  denoted  by  xi  and  X2,  then  all  possible  angles  are  given 
by  xi  ±  2  nir  and  x-z  ±2  nw. 

In  exceptional  cases  there  may  be  only  one  angle  <  360°,  as 
when  sin  a:  =  1  or  cos  x  =  —1. 

147.  Use  of  Tables  of  Natural  Functions.  —  Usually  the  angles 
corresponding  to  a  given  value  of  a  function  are  not  known 
exactly.     The  angles  may  then  be  found  approximately  by  the 


116 


TRIGONOMETRIC  FUNCTIONS 


[148 


aid  of  tables  of  the  natural  functions,  such  as  are  given  in  (125) 
and  in  Appendix,  Table  III. 

These  tables  give  the  functions  of  angles  from  0°  to  90°.  But 
they  will  serve  for  all  four  quadrants,  sirjce  any  function  of  any 
angle  is  reducible  to  a  function  of  an  acute  angle. 

When  the  given  value  of  the  function  is  not  found  exactly  in  the 
table,  the  corresponding  angle  must  be  obtained  by  interpolation. 
1 


Example  1.     Given  sin  x 


To  find 


The  two  values,  xi  and  Xi,  <  360°,  are  shown  in  the  figure.     They  are 


easily  found  when  xs,  the  angle  whose  sine  is  + 

xi  =  TT  +  2:3     and     X2  =  2ir 

Since  sin  0:3=  =  =  .333,  we  find  by  interpolation  from  Table  III,  X3  = 
19°  28'.     Hence,  xi  =  199°  28',    X2  =  340°  32'. 

All  possible  values  of  x  are  then  given  by 

199°  28'  ±  2mr,  340°  32'  ±  2mr. 
2 
Exam-pie  2.     Given  cot  -  x  =  3.362.     To  find  x. 

From  Table  III,  ^  x  =  16°  34'  or   196°  34'  ( =  180°  +  16°  34'). 

3  2 

Hence  all  possible  values  of  ^  x  are  given  by 

I  X  =  16°  34'  ±  2  riTT     or     196°  34'  ±  2  nir. 
Therefore,  x  =  24°  51'  ±  3  n^r     or     294°  51'  ±  3  mtt. 

We  might  also  write,  from  III  of  (146), 

I  X  =  16°  34'  +  wtt;  hence  x  =  24°  51'  +  \  nw. 

148.   Exercises.     Find  all  values  of  the  angles  which  satisfy 
the  following  equations: 

1.  cot  X  =  1 ;  sin  X  =  —  §;  sec  x  =  2;  cos  x  =  1. 

2.  esc  X  =  -V2;  tanx  =  VS;  cosx  =  .5;  cotx  =  —s/S. 

3.  sin  X  =  —5;  secx  =  —  3;  tanx  =  2;  cscx  =  5. 

4.  cosx  =  -.257;  cotx  =  -.998;  sinx  =  .020. 

5.  tan»  =  2.500;  csc0  =  -3.505;  sec  0  =  -10. 

6.  vers^  =  1.450;  vers^  =  .605;  covers  (/>  =  .750. 


149] 


TRIGONOMETRIC   FUNCTIONS 


117 


149.   Given  one  function  of  an  angle,  to  find  the  other  functions. 


Fiiui  the 


Example  1.     sinx  ■ 
other  functions.      ' 

Take  ordinate  =  1  and  distance  =  2; 
then  abscissa  =  db  V3  (figure). 
Then 


Wo 

cos  X  =  ±  jL2.  ,   tan  x  =  ± 
2 

cotx  =  ±V3, 

2 

sec  X  =  ±  —7Z. ,   CSC  X  =  2. 
V3 


V3 


We  have  found  two  values  for  each  function  except  csc  x,  which  is    the 
reciprocal  of  the  given  function.     Similar  results  will  be  found  in  general. 

Exam-pie  2. 


tan  X  = 


-3  +3> 

+4    ^---^y 


The  two  possible  positions  of  the  ter- 
X    minal  line  are  shown  in  the  figure. 

Hence,     sin  x  =  ± ->  cosx  =  ±  ~> 

4  ,5  5 

cot  X  =  —  ij  >  csc  X  =  ±  -  >  sec  X  =  ±  -7  • 


Example  3. 


Then  (figure), 


±2 
+3 


-^) 


Vl3 


VT3 


V3 


Vl3 


Example  4.       sinx=  -■ 

Ordinate  =  h;  distance  =  k; 
hence  abscissa  =  ±  v  A;^  —  K^. 


118  TRIGONOMETRIC   FUNCTIONS  •       [150,151 

Then  cos  a;  =  ±  -^^ ,   tan  x  =  ±  —r-  ,   etc. 

k  ■\lk'i  -  h^ 

Exercise  1.     Construct  figures  for  the  cases  when  ,  is  (a)  plus;  (b)  minus. 
Exercise  2.     Is  the  problem  possible  for  all  values  of  h  and  A;? 


Example  5.  tan  x  = 


fl  -6         /_  -  (fl  -b)\ 
2  \/ab        [       -  2  \/ab  ) 


Here  ordinate  =  a  —  b,  abscissa  =  2  V"'?! 

or,  ordinate  =  —  (a  —  6),  abscissa  =  —  2  Va6. 


In  either  case,  distance  =  +  y{a  —  6)''  +  4  a6  =  |  a  +  6 1 . 

TT  •  ,    fl  -  ft  ,2  -s/ab 

Hence,  smx  =  i  , r~r,'   cos  x  =  ±  , 1-^-,'  etc. 

|a  +  6|  \a  +  b\ 

Exercise  1.     Calculate  the  values  of  the  six  functions  when  a  =  2,  b  =3; 
when  a=  —  2,  6=  —  3;  when  a  =  1,  6  =  4;  a  =  —  1,  6  =  —  4. 
Exercise  2.     Is  the  problem  possible  for  all  values  of  a  and  6? 

150.   Exercises.     Find  the  other  functions,  given  that 

1.  sin  X  =  —  5.  6.  CSC  X  =  —  T  L  *^  «  ^ 

^  ^'  11.    CSC^  = 

2.  cosx  =  i  7.   secx  =-  |J-. 

12.  tan  9  =  a. 

3.  tanx  =  ^.  8.  cotx=-.75.  .,      .,       ,     * 

13.  sm  9  =  fl. 

4.  secx  =  4.  9.    sinx  =  .6.  14.    cot0  =  V^. 

5.  cotx  =  V3.  ^^  b  ^^  ,       a2  4-;^2 


10.    cos  0  =  -  •  15.   sec  (/>  = 


2ab 


16.    State  for  what  values  of  the  literal  quantities  in  exercises  10-15,  the 
given  equations  are  impossible. 

151.   To  express  all  the  functions  in  terms  of  one  of  them. 
1.    Express  all  the  functions  in  terms  of  the  cosine. 
We  have 

_  cosx  _  abscissa 
I  distance 

Hence  let      abscissa  =  cos  x  and  distance  =  1. 


Then  ordinate  =  ±  Vdist.^  —  absc-  =  ±  Vl  —  cos^  x. 


loll 


TRIGONOMETRIC   FUNCTIONS 


119 


The  figure  shows  this  graphically  when  cos  x  is  positive. 
Taking  into  account  both  values  of  tlie  ordinate,  we  have 


sina;=  ±  Vl-cos-a:; 
Vi  —  cos^  X 


tan  X  =  ± 
cot  .T  =  ± 

CSC  X  =  ± 


COS  a: 
COS  a: 


V 1  —  cos-  X ' 
1 

V 1  —  COS^  X 


sec  a; 


1 
cos  X 


Exercise  1.     Obtain  these  equations  for  the  case  when  cos  x  is  negative. 
Exercise  2.     Obtain   the  same  equations  directly  from   the   formulas   of 
Group  A. 

2.    Express  all  the  functions  in  terms  of  the  cotangent. 

cot  X      abcsissa 


cot  a;  = 


cot  a; 


-  1 


ordinate 


Hence  let  abscissa  =  cot  x  and  ord.nate  =  1  ; 

or  let  abscissa  =  —  cot  x  and  ordinate  =  —  1. 


In  either  case,  distance  =  +  Vl  -|-  cot^  x.     (See  figure,  where  we 
assume  cot  a;  >  0.) 


Hence     sin  a;  =  ± 


vl  +cot2a; 


cos  X  =  ± 


cot  a; 


Vl  +  cot-  X 


etc. 


120 


TRIGONOMETRIC   FUNCTIONS 


151 


By  taking  each  of  the  functions  in  turn,  and  proceeding  as 
above,  we  obtain  the  results  shown  in  the  fo  lowing  table.  The 
given  function  and  its  reciprocal  are  uniquely  determined;  the 
other  four  functions  are  ambiguous  in  sign. 


sin  X. 

cosx. 

tan  X. 

cot  X. 

sec  X. 

cscx. 

tanx 

1 

±Vsec2x-l 

1 

. 

±Vl-COs2x 

±VH-tan2x 
1 

±Vl+C0t2x 

cotx 

secx 

1 
secx 

cscx 

Vcsc2x-i 

±  Vl-sin2x 
sinx 

±Vl+tan2x 

±Vl+C0t2x 
1 

cotx 

CSCX 

±  Vl  — C0S2X 

1 

tanx 

±Vsec2x-l 

1 

±Vl-sin2x 

cosx 
cosx 

1 
tanx 

±VCSC2X-1 

itVl— sin^x 

cot  a: 

±VCSC2X-1 

sinx 

1 

±V1-C0S2X 

1 

cosx 

1 

±Vsec2x-l 

±Vl+C0t2x 

±Vl+tan2x 

±Vl-sin2x 

1 
sinx 

cotx 

secx 

±VCSC2X-1 

±Vl+tan2x 

±Vl+COt2x 

±Vl-COs2x 

tanx 

±Vsec2x-l 

Exercises. 

1.  Express  sin  x  cos2  x  +  sin^  x  in  terms  of  tan  x. 

2.  Express  tan  x  sec  x  +  sec2  x  in  terms  of  sin  x. 

3.  Express  cos2  x  tan  x  +  sin2  x  cot  x  in  terms  of  esc  x. 

in  terms  of  sec  d. 


4.    Express :; — ,     ,  _  „  + 


1  +  sm  ( 


_      ^  COS0  . 

6.   Express  ;; ytztt,  + 


1  -  sin  0 
sinff 


1  -  tan  0      1  -  cot  0 


in  terms  of  cos  d. 


CHAPTER   VIII 

Functions  of  Several  Angles 

152.  Formulas  for  sin  {oc  +  y)  and  cos  (x  +  2/).  —  Let  x  and  y 
be  two  angles,  each  of  which  we  first  assume  to  be  less  than  90°. 
Their  sum  will  then  fall  in  the  first  or  the  second  quadrant.  The 
two  cases  are  illustrated  in  the  figures,  and  the  demonstration 
which  follows  applies  to  either  figure. 

Construct  Z  XOP  =  x  and  Z  POQ  =  y,  the  terminal  side  of 
X  being  taken  as  the  initial  side  of  y. 


M  X 


From  Q,  any  point  on  the  terminal  side  of  y,  draw  perpendicu- 
lars NQ  and  PQ  to  the  sides  of  angle  x,  produced  if  necessary. 
Draw  MP  _L  OX  and  KP  _L  NQ. 

Then  Z  KQP  =  x,  and  in  either  figure. 


sin  (x  +  y) 


NQ      MP  +  KQ 


OQ 


OQ 


MP      KQ 

OQ  '^  OQ 


Hence 
(a) 


MP.    OP.    KQ    PQ 
OP  '  OQ.^  PQ^'  OQ^ 


sin  (ac  -\-  y)  =  sin  oc  cos  y  +  cos  x  sin  y. 

121 


122  FUNCTIONS   OF  SEVERAL  ANGLES  [153 

Also,  noting  that  ON  in  the  second  figure  is  a  negative  line, 

,     ,     .      ON      OM-NM      OM     KP 
cosix  +  y)  =  ^=         QQ         =-OQ-OQ 

^OM    OP__KP    PQ 
OP'OQ      PQ'OQ' 
Hence 

(b)  cos  (.-r  +  y)  =  cos  x  cos  y  —  sin  x  sin  ij. 

153.  In  the  above  proofs  we  have  assumed  x  and  y  less  than 
90°.     Similar  proofs  may  be  given  for  any  other  values  of  x  and  y. 

We  shall  however  use  formulas  (a)  and  (b)  to  verify  the  truth 
of  the  formulas 

(a')  sin  (A  +  5)  =  sin  A  cos  B  +  cos  A  sin  R, 

(W)  cos  (A  -\-  B)  =  cos  A  cos  B  —  sin  A  sin  B, 

for  all  values  of  A  and  B. 

A  and  B  will  differ  from  acute  angles  by  certain  integral  multi- 
ples of  90°,  say, 

A  =  X  +  w  .  90°;  B  =  y-^m-  90°. 

All  possible  quadrants  for  A  and  B  (except  the  first,  for  which  the 
formulas  have  been  derived)  will  be  included  by  considering  only 
the  values  1,  2,  3  for  w  and  m. 

In  particular,  let  n  =  1  and  w  =  2.     Then 

A=x-^  90°;  B  =  y+  180°;  A^B  =  x-\-y-^  270°. 

Hence,  if  formulas  (a')  and  (b')  are  true, 

'  sm{x  +  y-\-  270°)  =  sin  (x  +  90°)  cos  (y  +  180°) 

+  cos  (x  +  90°)  sin  (y  +  180°), 
cos  (x-\-y-\-  270°)  =  cos  (x  +  90°)  cos  (y  +  180°) 

-  sin  (x  +  90°)   sin  (y  +  180°). 

Removing  the  multiples  of  90°  by  the  rule  of  (139)  and  changing 
signs,  these  equations  reduce  to 

cos  (x  +  y)  =  cos  X  cosy  —  sin  x  sin  y, 
sin  (x  +  ?/)  =  sin  x  cosy  -\-  cos  x  sin  y. 


154,155]  FUNCTIONS  OF  SEVERAL  ANGLES  123 

But  these  are  true  since  x  and  y  are  acute  angles;  hence  also  (a') 
and  (b')  are  true.  In  exactly  the  same  way  the  truth  of  these 
equations  may  be  shown  for  any  integral  values  of  7i  and  m, 
positive  or  negative. 

Using  the  letters  x  and  y  in  place  of  A  and  B,  formulas  (a)  and 
(b)  are  true  for  all  values  of  x  and  y. 

154.  Replacing  y  by  —y  in  (a)  and  (b),  and  noting  that 

sin  (—?/)  =  —  sin y  and  cos  {—  y)=  cos  y,  we  have 

(c)  sin  (.K  —  y)  =  sin  a?  cos  y  —  cos  «  sin  y; 

(d)  cos  {x.  —  y)=  cos  oc  cos  y  +  sin  oc  sin  y. 

Equations  (a),  (b),  (c),  (d)  are  usually  called  the  addition  and 
subtraction  formulas  of  trigonometry.  All  the  other  working 
formulas  are  deduced  from  them.  - 

155.  Dividing  (a)  by  (b),  we  have 

,     ,      ,       sin  (x  +  v)       sin  x  cos  y  -f  cos  x  sin  y 

tan  {x -{-]))  = 7 — ,-^  = ^ -. ^^  • 

cos  {x  +  y)      cos  X  cos  y  —  smx  sm  y 

sin  X  cos  7/       cos  x  sin  y 


cos X cosy   '   cos X cosy 

sm  X  sm  y 
cos  X  cos  y 

ence, 

(e) 

.      ,     .       tan  .X  +  tan  y 

Similarly, 

(f) 

^  ,       ,       ,       cot  r  cot  V  —  1 

Also,  from 

(c)  and  (d),  by  division, 

(g) 

tan  .X  —  tan  y 

(h) 

cot(x-,v)  =  £2l£lM^. 

Exercises. 

1.   If  sin  X  =  J  and  sin  ?/  =  f,  calculate  sin  (x  -\-  y). 

(Four  answers:  ^  [±  Vo  ±  4^^].) 


12^  FUNCTIONS   OF  SEVERAL  ANGLES  [156 

2.  If  cos  a;  =  I  and  cos  y  =  |?,  calculate  cos  (x  +  y). 

3.  If  sin  a  =  3  and  sin  /3  =  |,  calculate  cos  (a  —  /3). 

Show  that, 

4.  cos  (60°  +x)  +  cos  (60°  -  x)  =  cos  x. 

5.  sin  (45°  +  0)  -  sin  (45°  -  6)  =  \f2  sin  5. 

.  «    ,   .      A       cos  (g  -  (f>) 

6.  cot  e  +  tan  0  =  -^^ — „   ^^  ,  • 

sin  0  cos  9 

7.  cos  (A  +  45°)  +  sin  (A  -  45°)  =  0. 

8.  sin  nd  cos  0  +  cos  nd  sin  0  =  sin;(n  +  1)  9. 

9.  tan^e-^)  +  cot(0+^)  =0. 

10.   From  the  functions  of  30°  and  45°  calculate  the  functions  of  75°. 

For  convenience  we  collect  formulas  (a),  (b)  .  .  .  ,  (h)  and  form 
Group  B,  numbering  them  consecutively  with  the  formulas  of 
Group  A. 

Formulas,  Group  B. 

(9)  sin  («  +  ?/)=  sin  xcosij  +  cos  oc  sin  p. 

(10)  cos  {x  +  2/)  =  cos  oc  cos  2/  —  sin  ao  sin  y. 

(11)  sin  («  —  2/)  =  sin  ic  cos  ?/ —  cos  ic  sin  2/. 

(12)  cos  (x  —  2/)  =  cos  X  cos  2/  +  sin  oc  sin  y. 

,     ,     s        tan  iT  +  tan  2/ 

(13)  tan(x  +  ,v)=^_^^^^^^,y 

,       cot  a?  cot  2/  —  1 

(14)  cot(x  +  2/)=    eotx  +  cot2/* 

tan  £r  —  tan  y 

(15)  tan(a.-2/)=^^tanxtan2.* 

.         cot  X  cot  2/  +  1 

(16)  cot(x-2/)=    ,ot^_eota.' 

156.  Functions  of  2  a?.  —  Putting  ?/  =  a;  in  (9),  (10),  and  (13)  of 
Group  B,  we  have 

(14)  sin  3  ic  =  2  sin  a?  cos  cc, 

(15)  cos  3  a?  =  cos'*  X  —  sin^aj, 

=  1  —  2  sin^  oc, 

=  2  cos^  a?  —  1. 

2  tana? 

(W)  tan  2  a?  = : — 5—  * 

^^^^  1  -  tan*  X 

For  cot  2  a:  use  - — ^  • 
tan2x 


157  ]  FUNCTIONS  OF  -SEVERAL  ANGLES  125 

Exercises. 

1.  Verify  these  formulas  when  x  is  30°;  45°;  150°;   -60°. 
Show  that, 

2.  2  CSC  2  X  =  sec  x  esc  x. 
1  -  tan2  X 


3.    cos  2  X 


1  +  tan2  X. 

sin  2  X 

=  tan  X. 


1  +  cos  2  X 
6.   tan  X  +  cot  X  =  2  esc  2  x. 

6.  Calculate  the  functions  of  2 x  when  sin x  =  1|. 

Ans.   sin  2x  =  ±||§;  cos2x  =  ^J^  ;  etc. 

7.  Calculate  the  functions  of  2  x  when  tan  x  =  \. 

157.   Functions  of  |  x.  —  The  second  and  third  values  of  cos  2  x 
in  (15)  are 

cos  2  a;  =  1  —  2  sin^  x, 
cos  2  a;  =  2cos2x  —  1. 


cos  2; 


Solving  these  for  sin  x  and  cos  x  respectively,  we  have 

,      /I  —  cos  2  X  ,  .  /l 

sin  a:  =  ±  y s »    cos  a:  =  ±  y  - 

Replacing  a;  by  i  x,  these  become 

(17)  sin|x=  ±y , 


(18)  cos|x=±y ^ 

Dividing  (17)  by  (18), 


,^f..       .      .  ,   . /l  —  cos  X      1— coscc 

(19)       tan  J  a?  =  ±  y  - 


+  cos  X  sin  a?  1  +  cos  x 

Formulas,  Group  C. 


(14)  sin  3 a;  =  2  sin  a?  cos  a?.        (17)  sin|a7  =  ±y  — 


cos  X 


/I  ox  1  ,   t  A  +  cos  X 

(18)  cos  |x  =  ±y 

(15)  cos  2  a?  =  cos' a?  —  sin' a?  '^ 

=  1  —  3  sin' as  /im  *      i  ,./l  — cosx 

(19)  tan^a?  =  ±Vrn 

=  2cos'x-l.  Vi  +  cosx 

_  1  —  cos  X 

sin  X 

,.  ^.   ,      „  3  tan  X.  sin  x 

(16)tan2x  =  - — J-.  =         — 

1  —  tan'  X  1  +  cos  X 


126  FUNCTIONS  OF  SEVERAL  ANGLES  [158 

Exercises. 

1.  Calculate  the  functions  of  15°  from  those  of  30°. 

2.  Calculate  the  functions  of  22|°  from  those  of  45°. 

3.  Calculate  the  functions  of  75°. 

4.  Calculate  the  values  of  tan  (2  x  -  y),  when  sin  x  =  I  and  cos  y  =  \i. 

Show  that, 

5.  sin  4  a;  =  2  sin  2  a:  cos  2  x.  ^^     1  +  sec  9  ^  ^  cos2  \  6. 

sec  0 
2  -  sec2  X 

6.  cos 2 X  =      ^^^2 X      '  13.   sin/3cot  J|3  =  1 +cos^. 


2  X  1  —  tan  X 


1  +  sin  2  X .      1  +  tan  x 

cos"  9  -  sin"  61  =  cos  2  6.  15.    cot  /3 

cos3  e  -  sin3  d       2  +  sin  2  0 


14.    1  +  tan  /3  tan  5  /3  =  sec  / 


2  cot? 


COS  0  —  sin  0  2  fl 

1  +  tan  ^ 

10.  cot  X  +  CSC  X  =  cot  I X.  16.    ,  ^"^^      = i. 

1  -  sm  /3      1  _  tan  ^ 

11.  (sin  1  0  +  cos  §  0)2  =  1  +  sin  0.  2 

158.   Formulas  for  sin  «  ±  sin  r  and  for  cos  u  ±  cos  f .  —  For- 
mulas (9)  and  (11)  of  Group  B  are 

sin  (x  +  y)  =  sin  xcosy  -\-  cos  x  sin  y, 
sin  {x  —  y)  =  sin  a:;  cos  ?/  —  cos  a;  sin  y. 

Adding:         sin  (x  +  y)  +  sin  (x  -  ?/)  =  2  sin  x  cos  2/. 

Subtracting:  sin  (x  -\-  y)  -  sin  (.r  -  ?/)  =  2  cos  x  sin  ?/. 

Let  X  -\-  y  =  u,     and       x  -  ?/  =  y  ; 

?^  +  y  ,  u  —  V 

then  a:  =  — ^       and     ?/  =      ^     ■ 

Substituting  in  the  two  preceding  equations,  we  liave 

.     if  +  r         n  —  r 

(20)  sin  u  +  sin  *'  =  3  sin  — - —  cos  — - — 

u  +  r    .     11  —  r 

(21)  sin  n  -  sm  v  =  2  cos      ^      sin     ^     • 


158]  FUNCTIONS   OF  SEVERAL  ANGLES  127 

Proceeding  similarly  with  formulas  (10)  and  (12)  of  Group  B, 
we  obtain, 

(22)  cos  u  +  cos  f  =  2  cos  — — -  cos 


3 

(23)  cos  n  —  cos  v  =  —  3  sm  —7; —  cm  — 

The  last  four  cciuations,  called  the  addition  theorems  of  trigo- 
nometry, we  collect  as  the 


Formulas,  Group  D. 


(20)  sm  u  +  sm  v  =  3  sm  — :; —  cos  — ^ —  • 


(21)  sm  a  —  sm  *'  ==  3  cos  — - —  sm — - — 


*/  +  V        11  —  V 
(22)  cos  u  +  cos  V  =  2  cos  — - —  cos  — -—  • 


(23)  cos  u  —  cos  V  =  —  2  sm   — - —  sm  — - —  • 


Example  1.     Show  that  ^!"^+^!°^  = ^- 

^  smx  —  smy  X  —  y 

tan-^— 

_    .    X  +  y        X  —  y 
2  sin  — i^  cos 


sin  X  +  sin  7/ 2 2 

sin  X  —  sin  w      _        x  -\-  y  .    x  — 
2  cos      '      sm  — ?r- 


.  tan   -^ 

,      X  +  y      .X  —  y  2 

=  tan  — ^  cot     _       = 

2  2  ,_  .r-^ 


r.  7    r.      C1L       XI.  i.   cos  75    +  cos  15°  /^ 

Example  2.     Show  that   ^^o"^ 1^6  =  -  V3. 

cos  75  —  cos  15 

cos  75°  +  cos  15°    2  cos  45°  cos  30°       *,-o  .or>o     y^ 

.^Fs-^ r?5  =  — .-,  ■  .go  ■  or^o  =  -  cot  4o  cot  30  =  -  v3. 

cos  75  —  cos  15    —  2  sm  45  sm  30 


128  FUNCTIONS  OF  SEVERAL  ANGLES  [159 

Exercises.      Show  that: 

1.  sin  3  a;  +  sin  5  a;  =  2  sin  4  x  cos  x. 

2.  sin  10  0  +  sin  6  5  =  2  sin  8  e  cos  2  9. 
■    3.   cos  2  x  +  cos  4  X  =  2  cos  3  X  cos  x. 

4.  sin  7  a  —  sin  5  a  =  2  cos  6  a  sin  a. 

5.  cos  4  e  —  cos  6  e  =  2  sin  5  d  sin  6. 

3  X        X 

6.  cos  x  +  cos  2  X  =  2  cos  —  cos  ^  • 

7.  sin  30°  +  sin  60°  =  V2  cos  15°. 

8.  sin  70°  -  sin  10°  =  cos  40°. 

9.  sin  5  X  cos  3  X  =  ^  (sin  8  x  +  sin  2  x). 
10.  2  cos  10°  sin  50°  =  sin  60°  +  sin  40°. 
.  ^  sin  A  +  sin  B       ,      A  +  B 

cos  A  +  cosB  2 

^_     sin9  +  sin30       ,      „. 

12.    — 1 5-;:  =  tan  2  d. 

cos  0  +  cos  3  0 

13.  2  cos  a  cos  /3  =  cos  {a  —  13)  +  cos  {a  +  p). 

14.  sin  4  0  sin  e  =  I  (cos  3  0  —  cos  5  0). 

15.  cos  8  X  —  cos  4  X  =  —  4  sin  2  x  sin  3  x  cos  3  x. 

16.  sin  (2  X  +  3  2/)  +  sin  (2  X  -  3  y)  =  2  sin  2  X  cos  3  y. 

159.   Exercises  involving  the  use  of  formulas  (1)  to  (23). 

1.  If  sin  X  =  t  and  sin  y  =  |,  find  the  value  of  sin  (x  +  y)  and  cos  (x  +  y) 
when  X  and  y  are  both  in  the  first  quadrant. 

2.  As  in  exercise  1,  when  x  and  y  are  both  in  the  second  quadrant. 

3.  If  cos  X  =  I  and  cos  y  =  |?,  calculate  sin  (x  +  y)  and  cos  (x  +  y)  when 
X  and  y  are  both  in  the  first  quadrant. 

4.  As  in  exercise  3,  when  x  and  y  are  both  in  the  fourth  quadrant. 
6.    If  sin  X  =  i  and  sin  2/  =  f ,  calculate  all  values  of  sin  (x±  y). 

6.  If  sin  a  =  i  and  sin  /3  =  §,  calculate  all  values  of  cos  (a  ±  0). 

7.  If  cos  a  =  f  and  cos  /3  =  |,  calculate  all  values  of  tan  (a  ±  /3). 

8.  Calculate  sin  {x  +  y  +  z)  when  sin  x  =  tV,  sin  y  =  ^j,  sin  2  =  5^,  and 
X,  y,  z  all  lie  in  the  first  quadrant. 

.9.   As  in  exercise  8,  when  x,  y,  z  all  lie  in  the  second  quadrant. 

10.  Calculate  cos  (x  +  y  +  z)  when  cos  x  =  |,  cos  ?/  =  if,  cos  z  =  H,  and 
X,  y,  z  all  lie  in  the  first  quadrant. 

11.  As  in  exercise  10,  when  x,  y,  2  all  lie  in  the  fourth  quadrant. 

12.  Calculate  tan  (x  +  y)  when  tan  x  =  1  and  cot  y  =  V3. 

13.  Calculate  all  values  of  sin  2{x  -y)  and  of  tan  {2x  -y)  when  tan  x  =  i 
and  tan  y  =  j\. 

14.  Calculate  all  values  of  cos  (a  +  /?)  when  tan  a  =  m  and  tan  ^  =  n. 

15.  Calculate  cot  (a  —  /3)  when  tan  a  ==  a  +  I  and  tan  /3  =  a  —  1. 

16.  Calculate  tan  (a  +  /8)  when  tan  a  =  — —r  and  tan  (8  =  ^ — xT" 

X  -f-  1  Z  X  "T  1 

17.  If  tan  a  =  f  and  tan  /3  =  j\,  calculate  tan  (2  a  +  /3). 


159]  FUNCTIONS   OF  SEVERAL  ANGLES  129 

18.  Calculate  sin  75°,  cos  75°,  and  tan  75°,  by  use  of  the  relation  (a)  75° 
=  -iy-;  (b)  75°  =  135°  -60°. 

19.  Calculate  the  functions  of  202^;  of  7^°. 
Prove  the  following  identities: 

20.  sin  X  sin  (y  —  z)  +  sin  y  sin  (z  —  x)+  sin  z  sin  (x  —  y)  =  0. 

21.  cos  X  sin  (y  —  z)+  cos  y  sin  (2  —  x)  +  cos  2  sin  (x  —  y)=  0. 

22.  cos  (x  +  y)  cos  {x  —  y)  +  sin  {y  +  2)  sin  (y—z)  —  cos  (x  +  2)  cos  (x  —  2)  =  0. 

23.  cos  (x  —  y  +  2)  =  cos  x  cos  y  cos  2  +  cos  x  sin  y  sin  2 

—  sin  X  cos  2/  sin  2  +  sin  x  sin  y  cos  z. 

24.  sin  3  X  =  3  sin  X  —  4  sin^  x.  t^  _.    cos  (a  +  /3) 


26.    cos  3  X  =  4  cos^  x  —  3  cos  x. 
3  tan  X  —  tan^  x 

1  -  3  tan2  X 
cot3  X  —  3  cot  X 


-  -     ^      o  3  tan  X  -  tan3  x  _. 

26.   tan  3  X  =  -^j 5- — s 31. 

1—3  tan2  X 


27.  cotSx 

28.  tan  4  9 


3  cot2  X  -  1 
4  tan  g  (1  -  tan2  g) 
1  -6tan2|9  +  tan4  9' 


sin  a  cos  /3 

=  cot  a  —  tan  p. 

cos  (a  -  /3) 
COS  a  sin  /3 

=  cot  /3  +  tan  a. 

sin  (a  —  0) 
COS  a  cos  /3 

=  tan  a  —  tan  0. 

sin  (x  +  y) 

tan  X  +  tan  y 

sin  (x  -  ?/) 

tan  X  —  tan  y 

cos  (x  +  y) 

_  cot  X  -  tan  2/ 

29.   '-^5j£_±^  =  tan«  +  tan^.  34.         ,  ,  .     ^. 

cos  a  cos  /3  cos  (x  —  y)       cot  x  +  tan  y 

35.  sin  (0  +  (f))  sin  {e  —  (p)=  cos2  ^  -  cos2  9. 

36.  cos  (w  4-  y)  cos  (w  —  v)=  cos2  u  —  sin^  v. 

37.  sin  (A  -  45°)  =  —  (sin  A  -cos  A). 

V2 


OQ         ,/,       ir\       cotA+1  on     +      /a     'r\       tanO-l 

38.    cot  ^A  -  ^ j  =  -^-^^  .  39.   tan  (^  -^j  =  ^^^^^^-^. 

40.   tan(^  +  .]  =  ^ 

41.    tan  fa +^j  +  tanf a  -  ^j  = 


tana 

cot  C5f 

cot2  a  -  3 ' 


.    5  7r  5  IT  49.   \/2sin(0  +  45°)  =  sin9  +  cos». 

sin  Y^r-  cos  — r 

42.    ^ 1^  =  2V3.  50.   sin2x  ^tanx 


51.   sec2x 


1  +  tan2  X 
csc^x 


43.   tanQ+9l= j^ r-  csc2x-2 


52.  cot  0  -  cot  2  fl  =  CSC  2  0. 

53.  sec2  «  COS  2  a  =  1  -  tan2  6. 

54.  1  +  tan  «  tan  2  a  =  sec  2  e. 

55.  1  —  COS  2  X  =  tan  x  sin  2  x. 
cot2  0  +  1 


56.   sec2( 


44.  cosfe+^Wsin(0-^)  =  O. 

45.  cot  (0  +  ^  W  tan  (^  -  ^)  =  0. 

46.  cotfe-^j  +  tan(0  +  ^)  =  O.  """   ''"^  "  "  "  cot2  0  -  1 

„         sin2(?  ^      „ 

47.  cot  I  -  tan  I  =  2.  67.    p^^^^  2-0  =  *^"  ^• 

48.  2cos^  =  V2  +  V2.  68.    ,  ^'"^^.,,  =  cot0. 

8  1  —  cos  2  e 


130  FUNCTIONS  OF  SEVERAL  ANGLES  [159 

59.  cot20-l  =2cot0cot2&.  _  1  -  cos 2 x 

60.  2  -  sec20  =  sec''0cos2e.  ^'^'   ^an  x  -  ^  _^  cos2a;' 

Ri         ^°^  2 ''  1  -  tan  g  cos  3  (?,  sin  3  9      ^     ^  ^ 

^^'    l+sin2g  =  r+^^-  64.    --j-^+___-=2cot2e. 

62.   '-^^  =  2COS2X  -  1.  65.   ^^"^  +  ^"^^  =  sec2.. 

cos  X  cot  9  -  tan  0 

66.   tan  (45°  +  <?i)  -  tan  (45°  -  0)  =  2  tan  2  0. 

g_    cos3  9!)  +  sin^  0  ^  2  —  sin  2  0 

cos  0  +  sin  0  2 

_-    cos5  (J)  —  sin^  (h       ,    .   ,    .    „         ,    .  „ 
68-  ^;;;ri -r— f  =  1 +  |sm2x- isin22x. 

cos  9  —  sin  9 

sin  X  +  cos  X       ,      ^  „ 

69.   ^ =  tan  2  x  —  sec  2  x. 

cos  X  —  sm  X 

4  tan2  X 


70.  sin  2  X  tan  2  X  = 


tan^x 

71.  cos2  0  +  sin2  0  cos  2  ^  =  cos2  (p  +  sin2  (^  cos  2  5. 

72.  1  +  cos  2  (i9  -  (;i)  cos  2  ^  =  cos2  d  +  cos2  {d  -  2  cjj). 

tan2f9  +  ^]-  1 

73.   ) i( =  sin20.    ■  75.    tanx  =       smx  +  sin2x      . 

tan2^  +  ^)+l  1+COSX  +  COS2X 


4/  ^       ,      ^  _«    .  sin  2  X  — sin  X 


COS  ( X  -\- 
74.   7 ^  =  sec2x— tan2x.         76.  tanx-, 

COs(x-^)  1-COSX  +  COS2X 

77.   sec25-Uan2gsin2g  =  ^5g^+4^. 
cot2  0  -  tan2g 

sin g  + cose  ^     /l+sin2g.         84.    i±-!5^  =  2cos2  ^. 
"sine-cose      yi-sin2e  sec  0  2 

79.  (sin  I  +  cos  ^)^  =  1  +  sin  e.  ^^-   ^^^~  ^  =  ^  tan  ?  esc  x. 

I     e  e\2  -.    1  +  cos  3  0        ^80!) 

80.  (^sin-  -  C0S2J  =  1  -  sine.  86.       ^j^^  3  ^       =  cot  —  - 

1  ^f      ^                      «7     1  +  sin  45°       ,       -_,„ 
1  +  tan  -  87.   .-0      =  tan  67=t°. 


81. 


1  —  sin  e      ,      ^      e  «^  1 

1  —  tan 


2  sec  e  +  tan 

tan  -  QQ     1  +  sin  x  +  cos  x 


o  o!'«    :; — ; — ;- —  =  COt  x 

i  =  «oo  ^  _  tor, -,.  l+sinx-cosx  2 


sec  X  —  tan  x. 


1  +  tan|  90_  t^^l  ^      /2^nx-sin2x. 

J.            J.  2       y/  2  sin  X  +  sin  2  X 

83.   tan  X  —  tan  p,  =  tan  X  sec  X.  ..  ,.-    .    „^„                „         ^ 

2            2  91.  V3  sin  75°  -  cos  75°  =  '\/2. 

no-5e      e      .9e      3e,       .     .  ^ 

92.  sm       cos     —  sin  —  cos  -  +  cos  4e  sin  2e  =  0. 

93.  sin  4  X  +  sin  2  X  =  2  sin  3  x  cos  x. 

94.  sin  3  X  +  sin  5  X  =  8  sin  x  cos2  x  cos  2  x. 


159]  FUNCTIONS   OF  SEVERAL  ANGLES  131 

-_     cot  15°  +  tan  15°        2  ^„      .     ,,,^„        .     ,^o        .    „^„ 

95.  ,  ,^o       , r^  =    — -•  97.    sin  100°  -  sui  40°  =  sin  20°. 

cot  15    —  tan  15        ^^3 

96.  ^~^^^"^'^°=-cotCO°.         98.    cos(j+a)+cos 
l-V2cos75°  V"^ 


100.    cos  (e  +  0)  -  sin  {d  -  <l>)  =  2  sin  f  ^  -  o)  cos  (^  -  (/.V 


V2. 


101.  2  sin  I  a  +  ,  j  sin  |  a  —  7  1  =  sin2  a  —  cos'^ . 

102.  sin  I  ^  +  a  j  —  sin  I  ^  —  a  I  =  V-  sin  a. 
V3-l_,,o 


103.   sin  40°  -  sin  10° 


V2 


tn.A      •    r,      .    •  .    -  n  ../v.-    Sin  75    +  sin  15  17, 

104.   sin  3  X  +  sm  x  =  4  sm  x  cos2x.         105.    ~. — ^^^^ -. — ^^^  =  y3. 

sin  75    —  sin  15 

^Qg^   COS  x  + cosy  ^  _  ^^^  X +^  ^^^  X- y 

cos  X  —  cos  y  2  2 

.„    sin  70°  +  sin  20°       .  ..„    sin  100°  + sin 40°       ..,  ,       _„ 

107.   -^o   , p^TT^  =  1.  108.    -■    ,„^ ■    .r>o  =  V3  tan  70 

cos70    +cos20  sin  100°  — sm  40 

109     (sin  «  +  sin  0)  (cos  a:  +  cos  0)  ^  _  ^^^2 

(sin  a  —  sin  0)  (cos  a  —  cos  0) 
j^^Q     (sin  g  +  sin  0)  (cos  a  -  cos  0)  ^  _  ^^^g 

(sin  a  —  sin  /3)  (cos  a  +  cos  0) 

111. 


(sin  75°  +  sin  15°)  (cos  75°  +  cos  15°) 
(sin  75°  —  sin  15°)  (cos  75°  —  cos  15°) 


.,  ^  _     cos  2  X  +  cos  12  X      cos  7  x  —  cos  3  x      ^  oi.*  ^  -^  _  /% 
cos  6  X  +  cos  8  X         cos  x  —  cos  3  x         sin  2  x 

113.  sin  X  +  sin  2  X  +  sin  3  X  =  4  cos  ^  x  cos  x  sin  |  x. 

(Hint.  Replace  sin  x  +  sin  3  x  by  2  sin  2  x  cos  x  and  sin  2  x  by  2  sin  x  cos  x ; 
from  these  results  factor  out  2  cos  x  and  combine  the  remainders  by  the  for- 
mula for  sin  u  +  sin  v.) 

114.  cos  X  +  cos  2  X  +  cos  3  X  =  4  cos  ^  X  cos  x  cos  i  x  —  1. 

115.  sin  2  X  +  sin  4  X  +  sin  6  X  =  4  cos  x  cos  2  x  sin  3  x. 

...     sine +sin20  +  sin3  0      ^      _„ 

116.    -^, K-^'r o-^  =  tan 2 9. 

cos6» +  cos2e  +  cos3  0 

117.  cos  20°  +  cos  100°  +  cos  140°  =  0. 

118.  cos  9  +  cos  3  0  +  cos  5  0  +  cos  7  0  =  4  cos  5  cos  2  0  cos  4  6. 

119.  sin  0  +  sin  3  0  +  sin  5  0  +  sin  7  0  =  16  sin  0  cos2  0  cos2  2  0. 

120.  4  sin2  f,6  cos2  </.  +  (cos2  ^5.  -  sin2  f/j)2  =  1. 

•    121.    (cos  xcosy  +  sin  x  sin  y)^  +  (sin  x  cos  y  —  cos  x  sin  y)-  =  1. 

.  „-      tan  3  X  —  tan  X        .      ^ 

122.    ,    ,   . — - — 7 =  tan  2  x. 

1  +  tan  3  X  tan  x 


123. 


tan  (n  +  1)0  —  tami 
1  +  tan  (n  +  1)0 tan; 


132  FUNCTIONS  OF  SEVERAL  ANGLES  [159 

124.  /r/'+/l"!r'^^=tang. 
1  +  tan  {e  +  ^)  tan  0 

^„,      tan  (0 —  </))+ tan  (i 

125.  :; f — ,,       ,,  , — —,  =  tan  e. 

1  —  tan  (fl  —  (j))  tan  ^ 

126.  sin  ne  cos  0  +  cos  nO  sin  &  =  sin  (n  +  1)  ^. 

127.  2  CSC  4  X  —  2  cot  4  X  =  cot  X  —  tan  x. 

128.  1^^«^  =  (1  +  cos  2  x)2.  129.   '^  -  '-^  =  2. 

1  —  cos  X  sin  0         cos  9 


2cosx 


130.  Iftanx  =-,  show  that  \/^-4  +  v/^-rT  =     , 

a  \  a-b      \  a  +  b      ^cos  2  x 

131.  4  cos3  X  sin  3  X  +  4  sin'  x  cos  3  x  =  3  sin  4  x. 

132.  sin3  X  +  sinS  (120°  +  x)  +  sin3  (240°  +  x)  =  -  f  sin  3  x. 

133.  cos  6  X  =  16  (cos6  x  —  sin^  x)  —  15  cos  2  x. 

134.  1  +  tan^  x  =  sec*  x  (sec2  x  —  3  sin2  x). 

^__    3  sin  X  —  sin  3 X 

135.  5 1 5-  =  tan3  x. 

3  cos  X  +  cos  3  X 

136.  sin  2  X  sin  2  2/  =  sin2  (x  +  y)  —  sin2  (x  —  y). 

137.  sin  5  a  sin  a  =  sin2  3  a  —  sin2  2  a. 

138.  cos*  a  =  ^  (3  +  4  cos  2  a  +  cos  4  a). 

139.  cos  2  X  +  cos  2  2/  +  cos  2  z  +  cos  2  (x  +  2/  +  2)  =  4  cos  (x  +  y)  cos  (y  +  z) 

cos{z  +  x). 

140.  sin2x  +  sin2y  +  sin2  3  +  sin2(x  +  y  +z)=  2  —  2cos  (x+?/)cos  (y  +  z) 

cos  (z  +  x). 

141.  cos2  X  +  cos2  y  +  cos2  z  +  cos2  (x  +  y  —  2)  =  2  +  2  cos  (x  +  y)  cos(x  —  2) 

cos  (y  —  2).  , 

^^'  /^  y         X  —  Z  V  "T~  2 

142.  sin  (x  —  y  —  z)  —  sin  X  —  sin  y  —  sin  z  =  4  sin  — ;^sin     _     sin  — ^ — 

143.  sin  2 a  +  sin 2/3  +  sin 2 7  =  sin  2  (a  +  /3  +  7)  +  4  sin  (a  +  ^)  sin (fi  +  y) 

sin  (a  +  7)- 

144.  sin  (a  +  /3  -  7)  +  sin  (a  -  /3  +  7)  +  sin  (/3  +  7  -  a)  -  sin  (a  +  /S  +  7) 

=  4  sin  a  sin  /3  sin  7. 

145.  cos  (a  +  /3  -  7)  +  cos  (/3  +  7  -  «)  +  cos  (a  +  7  -  /3)  -  cos  (a  +  /3  -}-  7) 

=  4  cos  a  cos  /3  cos  7. 

146.  Show  that  the  equation  sin  x  =  a  +  -  ib  impossible. 

147.  For  what  values  of  a  will  the  equation  2  cos  x  =  a  +  -  give  possible 
values  for  x  ? 

148.  Show  that  2  sin  =  =  —  Vl  +  sin  x  —  Vl  —  sin  x,  provided  that  x  lies 
in  the  second  or  third  quadrant. 

X 

2 
in  the  second  or  third  quadrant. 

150.   When  x  lies  in  the  fourth  quadrant,  show  that 


2  sin  2  =  Vl  -  sin  x  —  Vl  +  sin  x. 


CHAPTER   IX 

SliZ  X  tCLTi  jC 

Ratios- — ^and Inverse  Functions.       Trigonometric 

X  X 

Equations 

160.   The  limits  of  the  ratios  ^^^  and  ^^^-^  •     Let  a:  =  Z  NOP 

(figure)  lie  between  0°  and  90°;  let  NP  be  a  circular  arc  with  center 
at  0,  and  MP  and  NT  _L  ON.     Then 
MP  <NP  <  NT; 

MP      NP      NT 
hence         -qP^OP^OP' 

or         sin  X  <  x  (radians)  <  tan  x. 

That  is,  the  radian  measure  of  any 

acute  angle  lies  between  the  sine  and  the  tangent  of  the  angle. 

From  the  last  inequality  we  have,  on  dividing  l)y  sin  x, 

X 

1  <  -7^ —  <  sec  X. 
sm  X 

Suppose  X  to  decrease  and  approach  0.     Then  sec  x  =  I,  and  con- 
sequently also  ^^ —  =  1      and     -=  1. 

sui  X  X 

TT                                         ,.      sinsc 
-Hence  hm  =  1. 

Dividing  the  third  of  the  above  inequalities  by  tan  x,  we  have 

X 

cos  X  <  : <  1 ; 

tan  X 

letting  X  approach  zero  we  have 

,.      tan  X 
lim =  1 

x  =  0        ^ 

Hence,  the  ratio  of  either  the  sine  or  the  tangent  to  the  angle  (in 
radians)  approaches  1  as  its  limit  ivhen  the  angle  approaches  zero. 

133 


134  INVERSE  FUNCTIONS  [161 

When  angle  x  is  small,  these  ratios  will  be  nearly  equal  to  1; 

that  is, 

sin  X      .    ,              ,     tan  x      ^    , 
=  1  +  e     and     =  1  +  ei, 

X  X 

where  e  and  ei  are  small  quantities.     Hence 

sin  x  =  a:  +  ex    and    tan  x  ==  x  -{-  eix. 
Neglecting  the  small  terms  ex  and  eix,  we  have 

sin  X  =  tan  x  =  x  approximately,  when  x  is  small. 

Hence  when  x  is  small,  sin  x  and  tan  x  are  nearly  equal  to  x  {in 
radians) . 

The  degree  of  this  approximation  is  indicated  by  the  following 
values: 

Angle  X. 
Degrees  radians  sin  x  tan  x 

1°  .0174532925+  .0174524064+  .0174550649  + 

1'  .0002908882+  .0002908882+  .0002908882  + 

1"  .0000048481+  .0000048481+  .0000048481  + 

Exercises. 

1.  How  large  may  x  be  if  the  approximations 

sin  X  =  a;     and     tan  x  =  x 

are  to  be  correct  to  four  places  inclusive?     (Table.) 

2.  In  what  decimal  place  is  the  error  of  the  approximations 

sin  30°  =  30  sin  1°     and     tan  30°  =  30  tan  1°? 

3.  How  large  may  n  be  if  the  approximations 

sin  n°  =  n  sin  1°     and     tan  n°  =  n  tan  1° 

are  to  be  correct  to  three  decimals  inclusive  ? 

4.  As  in  exercise  3,  for  the  approximations 

sin  n'  =  n  sin  1'    and    tan  n'  =  n  tan  1'. 

161.  Inverse  Trigonometric  Functions.  —  It  is  often  convenient 
to  specify  an  angle,  not  by  its  degree  or  radian  measure,  but  by 
the  value  of  one  of  its  functions.  Thus  .we  may  speak  of  30°  as 
"an  angle  whose  sine  is  .\."  There  is  of  course  an  ambiguity 
here,  since  30°  is  only  one  of  the  angles  whose  sine  is  *. 


161]  INVERSE  FUNCTIONS  135 

If  X  is  an  angle  whose  sine  is  a,  we  write 

X  =  sin  - 1  a, 

which  may  be  read  "a:  equals  an  angle  whose  sine  is  a,"  or  "a; 
equals  the  inverse  sine  of  a,"  or  '' x  equals  anti-sine  a." 
Similarly  the  equation 

X  =  tan~^  a 

is  read  "a:  equals  an  angle  whose  tangent  is  a,"  or  "x  equals  the 
inverse  tangent  of  a,"  or  "a:  equals  anti-tangent  a,"  and  so  on  for 
the  other  functions. 

Obviously  the  equations 

X  =  sin  -  ^  a     and     sin  a:  =  o 
are  equivalent.     Similarly  for 

X  =  tan  -  ^  a    and    tan  x  =  a, 

X  =  sec ~^  a    and     sec  x  =  a, 
and  so  on. 

It  should  be  noted  that  "~^"  in  sin-^  a  is  not  an  exponent;  it 
might  equally  well  have  been  written  as  a  subscript,  sin  - 1  x,  or 
in  any  other  convenient  way.  The  reason  for  writing  it  as  above 
will  appear  by  noting  that,  according  to  the  laws  of  exponents, 
the  algebraic  equations 

X  =  h-'^a    and    bx  =  a 

are  equivalent. 

When  it  is  necessary  to  write  sin  x  with  an  exponent  —  1,  it 
should  be  written  (sin  x)-^  7iot  sin-^a:. 

The  smallest  positive  angle  whose  sine  is  a  is  often  called  the 
principal  value  of  the  symbol  sin-^a.  Similarly  for  the  other 
functions. 

If  6  denote  the  principal  value  of  any  inverse  function,  we  have 
from  (146),  equations  I,  II,  III,  • 

sin  -  1  a  =  2  nw  -\-  6,  or  esc  -  ^  a  =  2  rnr  +  9,  or 

(2/i  +  l)7r-0;  (2n  +  l)ir-d; 

cos- '^  a  =  2  riT  ±  6;y  sec^  a  =  2  mr  ±  6; 

tan -^  a  =  HTT  -^  6;  cot'^  a  =  nr  +  6. 


136  INVERSE  FUNCTIONS  [162 

162.  Equations  Involving  Inverse  Functions.  —  In  this  article 
we  shall  restrict  the  symbol  for  the  inverse  functions  to  mean 
only  the  principal  value  of  the  function.  Thus,  sin-  H  shall  mean 
the  angle  30°  only,  tan-  U  =  45°,  and  so  on. 

Exayn-ple  1.   Show  that  sin-i^  =  cos-i  ^• 

3  4 

Let  X  =  sin-i^    and    y  =  cos-i^; 

to  prove  that  x  =  y, 

or  that  sin  x  =  sin  y. 

(We  use  the  sine  for  convenience;  any  other  function  might  be  used.) 

.  3       .  .„_,3 

Now 


Also  cos?/  =  ^;    hence    sin  y  =  yl  —  cos  2/2  =  _.    q. e.d. 

4 
Example  2.   Show  that  2  tan-i2  =  sin-i  ^• 

4 
Let  X  =  tan- 12     and     y  =  sin-i^; 

to  prove  that  2x  =  y, 

or  that  sin  2  x  =  sin  y. 

Now  sin  2  X  =  2  sin  X  cos  x. 

2  1 

But  tanx  =  2;     hence    sin  x  =  — p    and     cosx==— p-    (149.) 

4 
Therefore  sin  2  x  =  -  =  sin  y.     q.  e.  d. 

Observe  that  if  x  were  not  restricted  to  be  the  principal  value  of  tan-i2, 

2 

we  might  have  sin  x  = p- 

V5 

2 
Example  3.    Show  that  tan-i  -  +  tan-i  2  +  tan-i  8  =  x. 

2 
Let  X  =  tan-i-;   y  =  tan-i2;   z  =  tan-i8; 

2 
then  tan  x  =  -  ;  tan  y  =  2;  tan  z  =  8. 

To  prove  that  x  +  y  +  z  =  n, 

or  that  x  +  y  =  w  —  z, 

or  that  tan  {x  +  y)  =  tan  (tt  —  2)  =  —  tan  z. 

XT  X       /     .      \        tan  X  +  tan  y        §  +  2  _  •  , 

Now     tan  (x  +  y)  =  -z .  .       ■  =  I j  =  -  8  =  -  tan  z.     q.  e.  d. 

"        1  —  tan  X  tan  y       1  —  | 

Example  4.   Show  that  tan-i  a  =  sin-i     ,  when  a  >  0. 

Vl  +  a2 

Let  X  =  tan- la    and    2/  =  sin- 


Vl  +  0.2 


then  tan  x  =  a    and    sin  y  = 


Vl  +a2 


163,  164]  TRIGONOMETRIC   EQUATIONS  137 

To  prove  that  ^  =  V, 

or  that  sin  x  =  sin  y. 

Now  since  x  and  y  stand  for  principal  values,  and  a  is  positive,  both  angles 
are  in  the  first  quadrant. 

Then  from  tan  x  =  a  we  find  (149) 

ffl 
sin  X  =     ,  _        , 
Vl+a2 
which  is  sin  y.     q.  e.  d. 

Discuss  the  above  example  when  the  symbol  for  the  inverse 
functions  is  assumed  to  stand  for  all  angles  having  the  function 
in  question,  instead  of  the  principal  value  only. 

163.    Exercises. 

1.    Show  that  the  equation  in  example  4  is  not  true  for  principal  values 
when  a  is  negative.     (Try  a  =  —  1.) 
Prove  the  following: 

,5,,        ,1      TT  6.    cos-ii+2sin-iA  =  120°. 

2.  tan-i^  +  tan-ig  =  ^.  ^    2  tan-i  3  =  sin-i  f. 

3.  2tan-ii=tan-i|.  g^    3  sin- 1  .^  =  sin- 1 -|  • 

4.  tan-i3+^  =  tan-i(-2).  ^    o  cot-i2  =  csc-i  |. 

5.  tan-i^  +  csc-i  VlO  =^-  10.    4tan-ii  =  tan-i^^g  +  |- 


2 

11.  tan- 1  ^  +  tan- 1 1  +  tan- 1  ( ^ 

,4   ,     .      ,  8    ,     .      ,13      TT 

12.  sm-i^  +  sm-i^  +  sm-ig^  =  2- 

13.  cos-i|i  +  2tan-4  =  sin-i|- 

65  5  5 

i>i     o.        ,2  ,5        .      ,33 

14.  2  tan- 1  -  —  esc- 1  -  =  sm- 1  —  • 

3  o  65 


')- 


15.    sin- 1  a  =  cos- 1  Vl  —  «^>  if  «  >  0- 

2w 


16.    2  tan- 1  ?«  =  tan- 1 


1  -m2 

cot2  e  -  tan2  (9 1 


17.    2  tan- 1  (cos  2  0)=  tan 

164.  Trigonometric  Equations.  —  A  trigonometric  equation  is 
an  equation  which  involves  one  or  more  trigonometric  functions  of 
one  or  more  angles.     Thus: 

sin^  x-\-  COQX  =  1 ;  tan  0  +  sec  ^  =  3;  cot  a  esc  a  =  2. 

To  find  the  values  of  the  angle  which  satisfy  such  an  equation, 
it  is  usually  best  to  use  a  method  adapted  to  the  case  in  hand. 
We  give  here  one  general  rule,  which  covers  a  considerable  variety 
of  cases. 


138  TRIGONOMETRIC  EQUATIONS  [165 

Rule:  To  solve  a  trigonometric  equation,  express  all  its  terms 
by  means  of  a  single  function;  solve  as  an  algebraic  equation,  con- 
sidering this  function  as  unknown;  find  the  angles  corresponding 
to  the  values  of  the  function  so  obtained.  Check  all  answers  hy 
substitution. 

Examples. 

1.  sin2  X  +  cos  X  =  1. 

Expressing  all  terms  by  means  of  cos  x,  we  have 

1  —  cos2  X  +  cos  X  =  1,     or     cos2  X  —  cos  X  =  0. 
cosx  =  0,     or     cosx  =  1. 

Hence  x  may  be  any  odd  multiple  of  |  or  any  multiple  of  2  7r;  i.e.,  if  n  be 
any  integer  or  zero, 

x=±(2n  +  l)^     or     x=±2mr. 

Exercise.     Check  these  answers  by  substitution. 

2.  tan0  +sec0  =  3. 

Expressing  all  terms  by  means  of  tan  d,  we  have 


tanO  ±  Vl  +tan2!?  =  3,  or  ±  Vr+tan2(?  =  3  _  tanff. 
Squaring  and  reducing, 

tan  0  =  ^ ;  hence     0  =  53°  8'  ±  nir. 
o 

When  n  is  odd,  these  values  of  e  do  not  satisfy  the  given  equation.     Hence  the 

solutions  are 

0  =  53°  8'  ±  2  nir. 

3.  cot  a  CSC  a  =  2. 
Then     ±  cot  a  Vl  +  cot2  «  =  2,  or  cot^  a  +  cot2  a  =  4. 


Hence  cot  a  =  ±  V—  |  ±  ^  Vl7. 

Using  the  upper  sign  under  the  radical  (the  lower  sign  makes  a  imaginary), 

we  have 

cota  =  ±  1.2496  + ;     hence     a  =  ±  38°  40'  ±  nir. 

When  n  is  odd,  the  values  of  a  must  be  discarded.     Hence 
a  =  ±  38°  40'  ±  2  nx. 
The  reason  for  the  additional  values  in  the  last  two  examples  is  that  in 
example  2  we  really  solved  both  the  equations  tan  e  ±  sec  6  =  Z,  and  in  exam- 
ple 3,  both  the  equations  cot  a  esc  a  =  ±  2. 

165.  Examples  Illustrating  Special  Methods.  —  These  depend 
chiefly  on  transforming  the  given  equation  by  means  of  some 
of  the  standard  formulas. 


165]  TRIGONOMETRIC  EQUATIONS  139 

4.  2  sin2  X  —  3  sin  x  cos  x  =  1. 

Since  2  sin2  x  =  1  —  cos  2  x     and     2  sin  x  cos  x  =  sin  2  x,  we  have 

3  9 

1  —  cos2x  -  -sin  2x  =  1,     or     tan2x=— ". 
^  6 

Hence  2x  =  tan-M  -  ^J  =  -  33°  41'  ±  titt. 

X  =  -  16°50'.5    ±  n|- 

Exercise.     Check  these  answers.     Solve  the  given  equation  by  expressing 
cos  X  in  terms  of  sin  x. 

5.  sin  3  v/  —  sin  2  f/  =  0. 

By  formula  (21)  of  (158)  this  becomes 

5.1^ 

2  cos  2  ?/sm~?/  =  0. 

5  1 

Hence  cos  ^  ?/  =  0     or     sin  -  ?/  =  0. 

^y  =  ±  {'^  n  ^r  \)\^    ov     ^y  =  ±nir. 
2/  =  ±  (2n  +  l)^,     or     y  =  ±2  nir. 

6.  cos  X  +  cos  3  X  +  cos  5  X  =  0. 

Since  cos  x  +  cos  5  x  =  2  cos  3  x  cos  2  x,  we  have 

2  cos  3  X  cos  2  X  +  cos  3  x  =  0,     or     cos  3  x  (2  cos  2  x  +  1)  =0, 

Hence  cos  3  x  =  0,     or     cos  2  x  =  —  -  ■ 

3x  =  ±(2n  +  l)2.     or     2x=±^^±2mr. 

X  =  ±  (2  n  +  1)  ^     or     ±  5  ±  n-n-. 

7.  tan  4  a  tan  5  a  =  1 . 

This  may  be  written  tan  4  a  =  cot  5  a.     But  when  the  tangent  of  an  angle 

A  equals  the  cotangent  of  an  angle  B,  A  +  B  must  be  an  odd  multiple  of  J 
Hence  4a4-5a=  ±  (2n  +  l)| 

a=±(2n  +  l)^- 

Here  a  is  any  odd  multiple  of  10°. 

Otherwise  thus:     tan  4  a  —  cot  5  a  =  0;  hence  — -; : — r — ■  =  0: 

cos  4  a       sm  0  a 

sin  4  a  sin  5  a  —  cos  4  a  cos  5  a  cos  9  or 

or  — 


cos  4  a  sin  5  a  cos  4  a  sin  5  a 

cos9a  =  0,     or     9a=±(2n  +  l)^. 


140  •      TRIGONOMETRIC  EQUATIONS  [165 

Exercise  1.  Check  these  answers.  Draw  figures  for  several  values  of  a  as 
10°,  30°,  50°,  70°.     Discuss  the  case  a  =  90°. 

Exercise  2.  In  example  7,  in  passing  from  the  first  equation  to  the  second 
we  divide  by  tan  5  a,  which  is  permissible  only  if  tan  5  a  ?^  0.  Justify  the 
division. 

Exercise  3.   Justify  the  division  by  cos  x  in  example  4. 

8.   a  sin  0  +  &  cos  B  =  c. 

We  might  reduce  to  sin  e  or  cos  e  and  proceed  according  to  the  rule  of  (164), 
A  method  much  preferred  in  practice  is  as  follows. 

In  place  of  a  and  h  introduce  two  new  constants  in  and  M  such  that 

(o  =  7/icosM,       ^  (m=V^M=;6^ 

w  •     Ttf      whence      \  ,,       .        ,  o 

/6=msmAf;  >M=tan-i-- 

The  given  equation  then  becomes 

m  (sin  0  cos  M  +  cos  6  sin  M)  =  C    or    sin  {d  +  M)  =  — . 

Hence  if  we  let  sin- 1  x  represent  all  angles  whose  sine  is  x, 

0  +  M  =sin-i  — ,     or     0  =  sin-i M. 

m  m 


Va2  +62  a 

Graphic  Solution.     As  an  example,  we  take  the  equation 
sin  2  0  +  sin  e  +  2  =  0. 

We  want  the  values  of  d  which  reduce  the  expression  sin  2  ^  + 
sin  0  +  ^  to  zero. 

Let  1/ =  sin2  0  + sin^  +  s- 

Calculate  y  for  a  series  of  values  of  ^,  as  0  =  0°,  10°,  20°,  .  .  .  , 
and  plot  the  points  (^,  y)  in  rectangular  coordinates.  The  result- 
ing curve  will  show  the  approximate  values  of  Q  for  which  y  is  zero. 
Any  convenient  scales  may  be  used  on  the  axes  of  Q  and  y. 

Let  the  student  read  off  the  required  solutions  from  the  graph 
below. 


166] 


TRIGONOMETRIC  EQUATIONS 


141 


Exercise.  By  means  of  this  graph  solve  the  equations 

(a)  sin  20  +  sin0=  0; 

(b)  sin20 +sin(?=  1; 

(c)  sin20  +  sin0=  \. 

166.   Exercises.     Solve  the  following  equations: 


1.  2  sin2  X  —  3  cos  x  =  0. 

2.  4  sin2  a  +  1  =  8  cos  a. 

3.  sin  a  +  cos  a  =  V2. 

4.  tan  e  +  coi9  =  2. 

5.  tan  /3  +  3  cot  /3  =  4. 

6.  2  sin2  X  +  3  =  5  sin  X. 

7.  2  (1  —  sine)  =  COS0. 

8.  5  sin  e  + 10  cos  0  =  11. 

9.  cos  2  X  =  cos^  X. 

10.  2  cos  2  X  =  1  +  2  sin  x. 

11.  4 cot 20  =  cot2  0-tan2  0. 

12.  cos  0  =  sin  2  d. 

13.  tan  2  X  =  3  tan  x. 

14.  sin  2  2/  =  cos  3  y. 

15.  tan  a  =  cot  3  a. 

16.  cot  8  ?!>  =  tan  ^. 

Solve  some  of  the  above  equations 

12,  13,  14,  15,  26,  28,  29. 


17. 
18. 
19. 
20. 

21. 

22. 
23. 
24. 
25. 
26. 
27. 
28. 
29 
30. 
graph 


sec  px  =  CSC  qx. 

tan  ?/  =  cot  6  y. 

sin  rd  =  sin  sd. 

cot  (30°  -  x)  =  tan  (30°  +  3  x). 


sin  4  a  =  cos  5  a.  _ 

sin(60°-x)-sin(60°+x)=iV3. 
sin  2  0  +  sin  4  0  =  V2  cos  d. 
sin(3O°+0)-cos(6O°+0)  =-.VV3. 
sin  4  a  =  cos  3  a  +  sin  2  a. 
sin  3  /3  +  sin  /3  =  cos  /3  —  cos  3  /3. 
sin  X  +  sin  2  X  +  sin  3  X  =  0. 
sin  X  +  sin  3  X  +  sin  5  X  =  0. 
cos  X  +  cos  2  X  =  008  i  x. 
ically,  in  particular  1,  2,  4,  5,  7,  8, 


142  TRIGONOMETRIC   EQUATIONS  [167 

167.  Simultaneous  Trigonometric  Equations.  —  We  shall  now 
give  some  examples  to  illustrate  methods  for  solving  a  system 
of  simultaneous  trigonometric  equations  for  several  unknown 
quantities.  To  express  answers  concisely,  we  shall  now  use  the 
symbols  for  the  inverse  functions  to  mean  all  the  angles  deter- 
mined by  the  given  function. 

Examples. 

1.    Solve  for  r  and  6:       r  cos0  =  x, 
r  sin  a  =  y. 
Squaring  and  adding,  r^  =  x^  +  y^; 


hence  r  =  ±  Vx2+t^. 

Divide  the  second  equation  by  the  first, 

tan  ^  =  -  ;     hence     6  =  tan- 1  - . 
a; '  .  X 

2.  Solve  for  a  and /3: 

a  sin  a  +  6  sin  (3  =  c, 

d  sin  a  +  e  sin  (S  =  /. 
Solve  for  sin  a  and  sin  /3  as  unknowns;  hence  get  a  and  0. 

Exercise.     Carry  out  the  solution  of  example  2.     Is  the  solution  possible 
for  all  values  of  a,  6,  .  .  .  ,/?     (62.) 

3.  Solve  for  r  and  d: 

dr  sin  0  +  &r  cos  0  =  c, 
a'r  sin  9  +  Vr  cos  6  =  c'. 
Solve  for  y  sin  6  and  y  cos  0  as  unknown;  then  proceed  as  in  example  1. 
Exercise.     Carry  out  the  solution  in  example  3. 

4.  Solve  for  X  and  2/ : 

y  =  sin  x, 
y  =  sm2x. 
Subtracting,  sin  2  x  —  sin  a;  =  0    or    2  sin  x  cos  x  —  sin  x  =  0. 
Hence  sin  x  =  0     or     cos  x  =  A. 

X  =  ±  nw     or     ±  60°  ±  2  mr. 
y  =  0     or     ±  i  V3. 
Exercise.     Solve  example  4  graphically. 

5.  Solve  for  y  and  <:  y  =  a  sin  (n<  +  &), 

y  =  a'  sin  {nl  +  b'). 

Equating  the  values  of  y,  and  expanding, 

a  (sin  ni  cos  b  +  cos  nt  sin  b)  =  a'  (sin  nt  cos  b'  +  cos  nt  sin  b'). 

Dividing  by  cos  nt  and  solving  for  tan  nt, 

a' sin  b'  —  a  sin  b 

tall  nt  =  r ; r,  • 

a  cos  0  —  o  cos  b' 

This  determines  a  set  of  values  of  nt.     Then  y  is  obtained  by  substituting 
in  either  of  the  given  equations. 


168]  TRIGONOMETRIC   EQUATIONS  143 

6.    Solve  for  r,  9,  and  f/>:  x  =  r  cos  6  cos  (f>, 

y  =  r  cosO  sin  0, 

z   =  r  sin  6. 

Dividing  the  second  equation  by  the  first,  we  have 

-  =  tan  0:     hence     6  =  tan-i -• 
X  ^'  '^  X 

Squaring  the  first  two  equations  and  adding, 

x2  4-  2/2  =  r2  cos2  0;     hence     r  cos  5  =  ±\Jx^  +  ?/2. 
Combining  this  result  with  the  third  equation,  as  in  example  1,  we  have 
tanO  = ;     hence    0  =  tan" 


±  Va;2  +  2/2 ' 

±  Va;2  +  2/2 

r2  = 

X2  +  2/2  +  22. 

168.   Exercises. 

Solve  for  r  and  »: 

1.   r  cos  0  =  3, 

r  sine  =  4. 

6.   rsin(e  +  ^)=2, 

2.   r  cos  0  =  12, 

rcos[e-^)=l. 

r  sin  9  =  —  5. 

7.   r  =  sin^e  +  ^), 

3.   r  cos  (?  =  -  9, 

r  sin  9  =  -  40. 
4.'  r  cos  9  +  2  7  sin  0  =  3, 

2  r  =  sin  f  e  -  ^]  • 
8.    r  =  2sin^2e-^j 
r  =  3sin(e  +  'f). 

rsinO  =  1. 

5.    r  (2  sin  0  +  3  cos  6)  =  1, 
r  (sine  +  4cose)=  1. 

Solve  f or  r,  6,  and  <P: 

9.   r  cos  e  cos  0  =  3, 

10.    r  cos  6  cos<l>  =  —  1, 

r  cos  0  sin  c/)  =  4, 

r  cose  sin<^  =  1, 

r  sin  e  =  5. 

r  sin  e  =  -  2. 

Eliminate  0  from  the  following  equations: 

11.  X  =  r  cos  6;  y  =  r  sin  e. 

12.  X  =  a  cos  e;  7/  =  6  sin  e. 

13.  X  =  a3  cos-''  9]  y  =  b^s'm^  9. 

14.  -cose  +  ^sine  =  1;  -  sin  e  -  |co3  e  =  -  1. 
a  0  a  0 

15.  Eliminate  9  and  ?!>  from  the  equations 

X  =  r  cos  9  cos  </>;  y  =  r  cos  e  sin  </>;  2  =  r  sin  e. 

16.  The  same  for  the  equations 

X  =  a  cos  9  cos  (!>',  y  =  b  cos  e  sin  ^;  z  =  c  sin  e. 


CHAPTER  X 

Oblique  Plane  Triangles 

169.  Between  the  six  parts  of  a  plane  triangle  there  exist, 
aside  from  the  angle-sum  equal  to  180°,  two  other  fundamental 
relations  which  we  proceed  to  obtain.  Additional  relations  will 
then  be  derived  from  these. 

The  Law  of  Sines.  —  In  any  plane  triangle,  the  sides  are  pro- 
portional to  the  sines  of  the  opposite  angles. 


Let  ABC  be  the  triangle,  CD  one  of  its  altitudes.     Two  cases 
arise,  according  as  D  falls  within  or  without  the  base  (figures). 
Then  in  the  first  figure, 

from  A  ACD,  h  =  bsmA; 

from  A  BCD,  h  =  asmB; 

equating  the  values  of  h, 

b  sin  A  =  asm  B,  or  a  :  6  =  sin  A  :  sin  B. 


In  the  second  figure, 
from  A  ACD, 
from  A  BCD, 


h  =  h  sin  (t  —  A)=  h  sin  A ; 
h  =  a  sm  B; 


equating  the  values  of  h,  we  find  the  same  result  as  before. 

144 


170,171]  OBLIQUE  PLANE  TRIANGLES  145 

By  drawing  perpendiculars .  from  the  other  vertices  and  com- 
bining results  we  have  the  Law  of  Sines, 

(1)  a:b.c  =  smA:sinB  :  sin  C. 

170.  The  Law  of  Cosines.  —  In  any  plane  triangle,  the  square 
of  any  side  equals  the  simi  of  the  squares  of  the  other  two  sides,  minus 
twice  their  product  by  the  cosine  of  their  included  angle. 

In  the  above  figures  let  AD  =  m.     Then 

First  figure.             '  -            Second  figure. 

in  A  AC  D,     m  =  b  cos  A;  m  =  b  cos(ir—A)  = —h  cos  A; 

inABCD,      a^  =  h'^-{-{c-m)2  a^  =  h^-\-{c-\-my^ 

=  h^+c^-2cm+m^  =h^-\-c^-^2cm  +  m~ 

=  62+c2-2cm.  =62+c2+2c?/^ 

Replacing  m  by  its  value  above,  we  have  in  either  case, 

(2)  tr  =  ft'  +  c'  —  2  be  cos  A. 
(2')  Similarly,  b^  =  tr -{■  c'  —  2  ac  cos  B. 
(2")                         jci  =  ff'  -\-b'*-2  (lb  cosC. 

171.  The  Law  of  Tangents.  —  In  any  plane  triangle,  the  dif- 
ference of  two  sides  is  to  their  sum  as  the  tangent  of  half  the  difference 
of  the  opposite  angles  is  to  the  tangent  of  half  their  sum. 

From  the  law  of  sines  we  have, 

a  _  sin  A 
b  ~  s'mB' 

By  composition  and  division,  and  subsequent  reduction  we  have, 
a  —  b      sin  A  —  sin  B 


a  +  b 

s'mA  +  sinjB 
_  2  cos  H^  +  B)  sin  i  (A  -  B) 

2  sin  \  {A  +  B)  cos  \  {A  -  B) 

=  cot  \  {A  +  B)  tan  h  {A  -  B). 

That  is, 

(3) 

a-b      tan  \  {A  -  B) 

146  OBLIQUE  PLANE  TRIANGLES  [172 

Similarly, 


(3') 


(3") 


«-c_tan|U-  C) 

a-j-c      tan  J  (^  +  C) ' 
&-  c  _  tan  3  (B  -  C) 

&  +  c~  tan|(^+  C) 


3 

The  symmetry  of  these  formulas  makes  them  easy  to  remember. 
In  actual  practice,  they  are  used  in  slightly  modified  form.  Thus 
the  first  of  them  is  written, 

tan^(A  -  B)  =  ^tan^(A  +5). 

Similarly  for  the  other  two. 

172.  Functions  of  the  Half -Angles.  —  When  the  three  sides  of 
a  triangle  are  known,  its  angles  are  best  calculated  by  the  formulas 
now  to  be  derived. 

From  the  law  of  cosines  we  have, 

62  +  c2  -  a2 


cos  A 


2  6c 


In  practice  this  formula  is  not  convenient  unless  a,  6,  and  c 
happen  to  be  small  numbers.     Now 


1^^^1-cosA,     (^^^^^^^^1-^.) 


sm^ 


But  1  —  cos  A 


1  - 

2  6, 

62  +  c2  - 

a2 

2  6c 
c  -  62  -  c2 

+ 

a^ 

0? 

2  6c 

-  (&  -  C)2 

(a 

2  6c 

+  6  -  c)  (a 

b+c) 

2  be 

Let  2s  =  a  +  6  +  c,     or     s  =  h  (a-\-h  -\-  c). 

Then     a  +  6  -  c  =  2  (s  -  c),     and    a  -  6 +  c  =  2  (s  -  6). 

rp,  ,  ,       4  (g  -  6)  (s  -  c) 

Then  1  -  cos  A  =  -^ —r^ > 

2  6c 

and 


(4)  .„1.  =  V/Il^^lll^. 


172]  OBLIQUE  PLANE  TRIANGLES  147 

Similarly,  

.    1           ,  /(s  -  a)  {s  -  c) 
(4')  sm-^  =  \/  ^^ 


(4")  sin 


1  „  _     /(s  -  a)  (s  -  />) 
3  ^  "  V  ah 


Observe  that  the  sides  appearing  explicitly  under  the  radical 
include  the  angle  to  be  calculated. 
To  obtain  cos  j  A,  we  have 


cos^A=y/^      , 

X  2  _1_  f,'. 

But  1+  cos  A  =  1  + 


2  6c 
(b  +  c)2  -  a2 

2  6c 
(6  +  c  +  g)  (6  +  c  -  g) 

2  6c 
4  g  (g.  —  a) 
2  he       ' 


Hence 


1   .  /s(s  -  a) 

V        6c 


(5)  cos-^  =  V^^ 

Similarly, 


2  V        ac 


(50  cos^ 

1  ,  /s  (s  —  c) 

(5")  "=5C  =  V^^- 

Dividing  sine  by  cosine  we  have 


(6)  tanl.^s/il^ffj^- 

(6")  tanlc  =  v/(l^^^i^. 


If 


_  J(s  -  <i)  (s  -  b)  (s  -  c) 


148  OBLIQUE  PLANE  TRIANGLES  [173,174 

then 

(7)  tani^ 


2  s  —  a 

(7")  tanic  =  ;^^. 

All  these  formulas  for  the  half-angle  should  be  memorized, 
preferably  in  verbal  form,  so  that  a  single  statement  contains  all 
three  formulas  of  any  one  set. 

173.  Solution  of  Plane  Oblique  Triangles.  —  A  triangle  is  deter- 
mined, except  in  such  cases  as  will  be  specially  mentioned,  when 
three  parts  are  given,  of  which  one  at  least  must  be  a  side.  ,The 
calculation  of  the  other  parts  is  called  ''solving  the  triangle." 

Four  cases  arise,  according  to  the  nature  of  the  given  parts. 

I.  Given  two  angles  and  one  side. 

II.  Given  two  sides  and  their  included  angle. 

III.  Give7i  two  sides  and  an  opposite  angle. 

IV.  Given  three  sides. 

The  method  for  treating  each  case  will  now  be  considered. 

174.  Case  I.     Given  two  angles  and  one  side,  as  A,  By  a. 
Formulas  for  finding  the  other  parts,  C,  h,  c. 

C  =  180°-  {A  +B). 

From  the  law  of  sines, 

sin  B  sin  C 

0  =  a- — 7 ;   c  =  a  -. — r. 
sm  A  sm  A 

Check.  It  is  important  to  have  a  check  on  the  accuracy  of  the 
calculated  parts.  For  this  purpose  use  any  formula  involving  as 
many  as  possible  of  these  parts. 

In  this  case  we  use 

6      sin  B  7    •    /-*  •     D 

-  =  -; — 7= ,     or    6  sm  C  =  c  sm  B. 
c      sm  C 

Example.     Given  A  =  50°,  B  =  60°,  a  =  150. 
To  find  C,  b,  and  c. 


174]  OBLIQUE   PLANE  TRIANGLES  149 

Solution  by  Natural  Functions. 

C  =  180°  -  (50°  +  60°)  =  70°. 

=  169.58. 


-          sinS 
h  =  a- — T  = 

sin  A 

150  X  .8660 
.7660 

sin  C 
c  =  a- — 7-  = 
sin  A 

150  X  .9397 
.7660 

6sinC  = 

c  sin  B, 

169.58  X  .9397  = 

184.01  X  .8660, 

159.35  = 

159.35. 

184.01. 


Check, 
or 
or 

Logarithmic  Solution. 

C=  180-  {A  -j-B). 

h  =  a  -. — J  ;   log  6  =  log  a  +  log  sin  B  +  colog  sin  A. 
sin  Jx 

c  =  a  -. — J  ;   log  c  =  log  a  +  log  sin  C  +  colog  sin  A. 
sin  A 

Check,     b  sin  C  =  c  sin  B;  log  b  +  log  sin  C  =  log  c  +  log  sin  B. 
We  now  write  out  the  following  scheme: 


A-\-B  = 

C  =  180° 

-{A-{-B)  = 

log  a  = 

log  a  = 

log  sin  B  = 

log  sin  C  = 

colog  sin  A  = 

log  6  = 

colog  sin  A  = 
logc  = 

6  = 

c  = 

Check.          log  b  = 

logc  = 

log  sin  C  = 

log  sin  B  = 

Now  turn  to  the  tables  and  take  out  all  the  logarithms  required, 
inserting  them  in  their  proper  places.  Add  to  obtain  log  b  and 
log  c.  Insert  these  in  the  check  and  add.  If  the  sums  in  the 
check  agree,  or  differ  by  only  a  unit  in  the  last  figure,  the  numerical 
work  is  correct.     Then  look  up  b  and  c. 


150 


OBLIQUE   PLANE   TRIANGLES 


[174 


On  making  these  calculations  with  the  data  in  our  example  the 
scheme  appears  as  below. 


A+B  =  110°. 

log  a  =  2.1761 

log  sin  B  =  9.9375 

colog  sin  A  =  0.1157 

log  h  =  ^2293 

h  =    169.6 

Check.        log  h  =  2.2293 

log  sin  C  =  9.9730 

2.2023 


C  =  180°  -  110°  =  70° 

log  a  =  2.1761 

log  sin  C  =  9.9730 

colog  sin  A  =  0.1157 

log  c  =  2.2648 

c  =    184.0 

log  c  =  2.2648 

log  sin  B  =  9.9375 

2.2023 


Remark.  In  calculating  with  four-place  logarithms,  three  sig- 
nificant figures  of  the  resulting  numbers  are  usually  correct.  The 
fourth  figure  should  be  retained,  but  may  be  one  or  more  units  in 
error.  It  is  rarely  worth  while  to  retain  more  than  four  significant 
figures. 

A  similar  remark  applies  to  5-,  6-,   and  7-place  tables. 


See 


chapter  on  numerical  computation. 

Graphic  Solution  of  Case  I ;  given 
A,  B,  and  a. 

Calculate  C  =  180°  -  {A  +  B). 
Lay  off  a  line  segment  equal  to  a 
and  at  its  extremities  construct  an- 
gles B  and  C,  prolonging  their  free 
sides  until  they  meet  at  A  (figure). 
Scale  off  the  lengths  of  b  and  c.  The 
figure  shows  the  triangle  already 
^  solved  above.    From  it  we  have 

b  =  167,  c  =  181. 


No  solution  is  possible  when  A  -\-  B  >  180' 


Exercises. 

solutions. 


Solve   the   following    triangles,    including    graphic 


1.  A  =  55° 

2.  A  =  6.5°  25' 

3.  C  =  34°  48' 

4.  B  =  115°  10'.  5 

5.  J3  =  88°  20' 


B  =  72° 
B  =  78°  23' 
A  =  100°  17' 
C  =  40°22'.3 
C  =  105°  30' 


a  =  1000. 
a  =  4.245. 
6  =  0. 5575. 
c  =  0.00275. 
a  =  10. 


175] 


OBJ.IQUE  PLANE  TRIANGLES 


151 


175.  Case  II.  Given  two  sides  and  the  included  angle,  as 
a,  b,  C. 

To  solve  the  triangle  we  calculate  H^  +  ^)  as  the  comple- 
ment of  AC;  then  h  {A  -  B)  is  calculated  by  formula  (3).  Angles 
A  and  B  are  then  determined  and  hence  all  the  angles  are  known. 
We  can  then  compute  c  in  two  ways  by  means  of  the  law  of  sines. 
The  agreement  of  the  two  values  of  c  furnishes  a  check  on  the 
computations. 

Formulas. 


HA+B)=90°-hC 

> 

tan  HA-B)=  ~~  tan  h  (A -\- B), 

sin  C 

^sinC 

c  =  a-. — J  = 
smA 

^inJ5- 

Scheme  for  Logarithmic  Solution. 

a  =                         log  (a  -  6)  = 

HA+B)  = 

h  =                      colog  (a  +  6)  = 

HA-B)  = 

a  +  b=             log  tan  \  (A  +  5)  =^ 

A  = 

a-b=             log  tan  A  {A  -  B)  = 

B  = 

log  a  = 

log  6  = 

log  sin  C  = 

log  sin  C  = 

colog  sin  A  = 

colog  sin  B  = 

logc  = 

logc  = 

c  =  c  = 

Graphic  Solution.  Construct 
angle  C  and  on  its  sides  lay 
off  lengths  a  and  b,  starting 
from  the  vertex.  Complete 
the  triangle,  and  measure  c,  A, 
and  B  (figure,  constructed  for 
example  below). 

A  solution  is  possible  pro- 
vided C  <  180°.  -"^ 

Example.    Given  b  =  12.55,  a  =  20.63,  C  =  27°  24'.    Solve  the 
triangle. 


152 


OBLIQUE  PLANE  TRIANGLES 


[176 


Logarithmic  Solution. 

HA+B)=90°-hC  =  90°-  13°  42'  =  76°  18'. 
a  =  20.63  log  (a  -  6)  =  0.9074     HA-\-B)=   76°  18' 

b  =  12.55  eolog  (a  +  6)  =  8.4792     ^iA-B)=  44°  58'.4 

a  +  6  =  33.18     log  tan  H^  +  g)=  0.6130  A  =131°  16'.4 

a-b=    8.08     log  tan  h{A-B)=  9.9996  B  =  31°  19'.6 

1.3145 
9.6630 
0.0682 


log  a  = 

log  sin  C  = 

cologsinA  = 

logc  = 


B)  =  9.9996 

log  b  = 

log  sin  C  = 

cologsin5  = 

logc  = 


A  =131' 
B  =    31' 

1.0986 
9.6630 
0.2841 
1.0457 


1.0457 

c  =  11.11  c  =  11.11 

Graphic  Solution.     This  is  shown  in  the  figure  above, 
student  scale  off  the  known  parts. 

Exercises.     Solve  the  following  triangles: 


Let  the 


1.  a  =  1500, 

2.  6  =  15.25, 

3.  a  =  1.002, 

4.  &  =  6238, 

5.  a  =  16.21, 


b  =  750, 
c  =  12.65, 
b  =  0.8656, 
c  =  4812, 
c  =  22.48, 


C  =  58°. 
A  =  98°  40'. 
C  =  130°  48', 
A  =  75°  22'. 
5  =  36°  54'. 


176.  Case  III.  Given  two  sides  and  an  opposite  angle,  as 
a,  6,  A, 

This  is  known  as  the  ambiguous  case.  We  begin  by  studying  the 
graphic  solution. 

Lay  off  angle  A  and  on  one  of  its  sides  take  AC=  b.  With  C  as 
center  and  radius  equal  to  a,  strike  an  arc  of  a  "circle.  The  figures 
show  the  various  possibilities  arising  in  the  construction,  the  first 
three  for  A  <  90°,  the  last  three  for  A  >  90°. 


176]  OBLIQUE  PLANE  TRIANGLES  153 

In  each  case  the  perpendicular  from  C  on  the  other  side  of  angle 
A  is  equal  to  6  sin  A.     Inspection  of  the  figures  then  shows  that 

when  A  <  90°  and  a  <  6  sin  ^,  no  triangle  is  possible; 

when  A  <  90°  and  a  =  b  sin  A,  a  right  triangle  results; 

when  A  <  90°  and  b  >  a  >  b  sm  A,  two  oblique  triangles  result; 

when  A  <  90°  and  a  >  b,  one  oblique  triangle  results; 

when  A  >  90°  and  a  =  6,  no  solution  is  possible; 

when  A  >  90°  and  a  >  b,  one  oblique  triangle  results. 

It  is  always  possible  therefore  to  state  in  advance  what  the 
nature  of  the  solution  in  a  given  case  will  be. 


Formulas.     Given  a,b,  A. 

f  .    „      b   .     .       (C  =  180°-(A  +  5). 
sinB  =  -sniA.  „       '  ^,. 

B'  =  180°-5. 


sin  C      ,  sin  C 
c  =  a- — 7  =  b- — ^' 
smA         sin  is 

,        sin  C      ,  sin  C 

c  =  a  -. — T-  =  0-. — ^ 

sin  A         sin  B 


Check.  The  agreement  of  the  values  of  c  and  c'  as  calculated 
from  the  two  expressions  for  each  of  them  furnishes  a  partial 
check  on  the  calculations.  It  does  not  guard  against  an  error  in 
log  sin  C,  which  may  be  checked  independently.  A  complete 
check  is  furnished  by  (6)  of  (172). 

In  carrying  out  the  calculations  according  to  the  formulas  above, 
the  various  cases  shown  in  the  figures  are  indicated  as  follows: 

(a)  log  sin  B  =  0;  no  solution,  or  right  triangle. 

(b)  retain  both  B  and  B';  two  solutions. 

(c)  A  +  B'  >  180°,  hence  reject  B';  one  solution. 

(d)  log  sin  B  =0;  no  solution. 

(e)  A  -\-  B  >  180°  and  A  -\-  B'  >  180°;  no  solution. 

(f)  As  in  (c);  one  solution. 

In  a  given  numerical  example  the  nature  of  the  solution  always 
becomes  apparent  during  the  progress  of  the  computations. 


154 


OBLIQUE  PLANE  TRIANGLES 


[177 


Example.     Given  a  =  602.3,  b 
Logarithmic  Solution.* 

log6  =  2.88316  loga  = 

colog  a  =  7.22019  log  sin  C  = 

log  sin  A  =  9.79217      colog  sin  A  = 

log  sin  S  =  9.89552  logc  = 

B  =    51°  50'.0  c  = 

5'  =  138°10'.0  loga  = 

A+B  =    90°    7'.3     logsinC'  = 

A+B'  =  166°27'.3  colog  sin  ^  = 

C  =    89°  52 '.7  log  c'  = 

C  =    13°  33'.7  c'  = 


=  764.1,  A  =  38°  17'.3. 


2.77981  log6  =  2.88316 

0.00000        log  sin  C  =  0.00000 
0.20783    colog  sin  B  =  0. 10448 


2.98764 

logc 

=  2.98764 

971.9 

2.77981 

logb 

=  2.88316 

9.36960 

log  sin  C 

=  9.36960 

0.20783  colog  sin  5' 

=  0.10448 

2.35724  log  c'  =  2.35724 

237.6 


Graphic  Solution.     This  is  shown  in  the  figure,  from  which  the 
unknown  parts  may  be  scaled  off. 

Exercises.     Solve  the  triangles  whose 
given  parts  are: 


1. 

a  =  29.95,              6  =  37.17, 

A  =  42°  24'. 

2. 

a  =  1756,                h  =  745, 

A  =  67°  30'. 

3. 

h  =  .  728,                c  =  .  542, 

B  =  105°  44'. 

4. 

h  =  6.174,               a  =  2.614, 

B  =  32°  22'. 

177.  Case  IV.     Given  the  three  sides,  a,  b,  c. 

The  angles  may  be  calculated  from  either  the  sine,  cosine,  or 
tangent  of  the  half-angles.  When  all  three  angles  are  wanted, 
it  is  best  to  use  the  tangent.  There  is  no  solution  when  one  side 
equals  or  exceeds  the  sum  of  the  other  two. 

Formulas. 


s  =  2  (a  +  &  +  c) 


;     r  =  \/ 


(s  —  a)  (s  —  b)  (s  —  c) 


tan  2  A 


tan- S 


r  1 

-3^;     tan^C 


Check.     HA-h  B  +  C)=90° 


*  The  fifth  figure  is  carried  to  avoid  accumulation  of  error.     This  is  advis- 
able if  all  possible  accuracy  is  desired. 


178] 


OBLIQUE  PLANE  TRIANGLES 


155 


Example.     Given  a  = 
Logarithmic  Solution. 

a    428.6 

h     806.2 

c     542.4 


428.6,  h  =  806.2,  c  =  542.4. 

cologs  7.0513 
log  (s  -  a)  2.6628 
log(s  -  h)   1.9159 


l_A  14°47'.7 
A  B  55°  51 '.5 
\  C  19°  20'.5 


2  s  1777.2 

log  (s  -  c)  2.5393 

Check  89°59'.7 

s    888.6 
s-a     460.0 

2|4.1693 
log  r  2.0846 

A     29°  35'.4 
S   111°43'.0 

s  -  b      82.4 

log  tan  *^  9.4218 
log  tan  ^B  0.1687 

C     38°  41'.0 

s  -  c     346.2 

179°  59'.4 

Check  1777.2  log  tan  iC  9.5453 

Graphic  Solution.     This  is  shown  in  the  figure, 
we  find  A  =  29°,  B  =  112°,  C  =  38°. 

B 


By  measuring 


ScaU 

Exercises.     Solve  the  triangles  whose  given  parts  are: 
.1.    a  =  6192,  b  =  4223,  c  =  7415. 

2.  a  =  156.21,  b  =  300.15,  c  =  410.32. 

3.  a  =  0.00245,  6  =  0.00405,  c  =  0.00536. 

4.  a  =  52.76,  6  =  22.84,  c=  28.41. 

178.  Areas  of  Oblique  Plane  Triangles.  — Referring  to  the  fig- 
ures of  (169),  we  see  that  h  is  the  altitude  drawn  on  side  c  as  base. 
Hence  if  K  denote  the  area  of  the  triangle,  we  have 

(8)  X  =  Wic  =  I  ac  sin  li.  (h  =  a  sin  b.) 

Hence,  the  area  of  a  plane  triangle  equals  half  the  product  of 

two  sides  by  the  sine  of  their  included  angle. 

The  area  is  also  expressible  in  simple  form  in  terms  of  the  sides. 

In  the  formula  above  replace  sin  B  by  2  sin  I  B  cos  h  B.     Then 

K  =  ac  sin  h  B  cos  ->  B 


=  acy 


a)(s 


_£_)  ./sji 


b) 


156 


OBLIQUE  PLANE  TRIANGLES 


[179 


by  (4')  and  (5')  of  (172).     Hence, 

(9)  JS:  =  Vs  (s  -  a){s  -  b){s  -  c). 

When  the  given  parts  of  the  triangle  are  such  that  neither  of  the 
above  formulas  applies  directly,  it  is  usually  best  to  calculate 
additional  parts  so  that  one  of  these  formulas  may  be  used. 

179.  Exercises  and  Problems. 


1. 

a  =  183.9, 
b  =  584.9, 
c  =  166.6. 

5. 

a  =  183.7, 
A   =36°  55'.  9, 
C  =70°  58'.  2. 

2. 

a  =  1.925, 
b  =  2.243, 
c  =  7.25. 

6. 

a  =283.6, 
A  =ir  15', 
B   =  47°  12'. 

3. 

a  =  42.31, 
6  =  71.70, 
c  =  71.35. 

7.  ^ 
a  =  783, 
B  =  42°  27', 
C   =  55°  41'. 

4. 

a  =  .41409, 
6  =  . 49935, 
c  =  .18182. 

8. 

c  =  22.504, 
B  =  55°  11', 
C  =  45°  34'. 

9. 

b  =  3069, 
B  =  15°  51', 
A  =  58°  10'. 

10. 

b  =  100.2, 
B  =  48°  59', 
C  =  76°  3'. 

11. 

a  =  3186, 
b  =  17156, 
A  =  147°  12'. 

12. 

a  =   .8712, 
b  =  .4812, 
A  =  24°  31'. 

13. 

a  =  1523, 
b  =  1891, 
A   =  21°  21'. 

14. 

A  =  61°  16', 
a  =  95.12, 
b  =   127.52. 

15. 

a  =  .39363, 
c  =  .23655, 
C  =  22°  32'. 

16. 

6  =  147.26, 
c  =  109.71, 
A  =  41°  15'. 

17. 

b  =  .5863, 
a  =  .8073, 
C  =  58°  47'. 

18. 

a  =  10.374, 
c  =9.998, 
J5  =  49°  50'. 

19. 

b  =  6.4082, 
c  =  18.406, 
A   =  33°  31'. 

20. 

b   =  .8869,^ 
a   =  3.0285, 
C  =  128°  7'. 

21. 

a  =  .8706, 
b   =  .0916, 
c  =  .  7902. 

22. 

a  =  20.71, 
b  =  18.87, 
C  =  55°  12'  3". 

23. 

A  =  41°  13', 
o  =  77.04, 
b   =  91.06. 

24. 

a  =  4663, 
&  =  4075, 
C  =  58°. 

25. 

a  =  43031, 
c  =  31788, 
A   =  19°  12'.  7. 

26. 

a  =  16082, 
c  =  13542, 
C   =  52°  24'.  3. 

27. 

a  =  .00502, 
&  =  .00558, 
c  =  .00466. 

28. 

b  =  2584, 
c  =  5726, 
A  =  27°  13'. 

29. 

b  =  37403, 
a  =  49369, 
A  =  81°  47'. 

30. 

a  =  6148, 
c  =  7512, 
A   =  133°  30'. 

31. 

a  =   .01520, 
fe  =  .03366, 
c   =  .02114. 

32. 

b   =  8204, 
c  =  9098, 
A  =62°9'.e 

179] 


OBLIQUE   PLANE  TRIANGLES 


167 


33. 

34. 

35. 

36. 

a   =  532, 

a  =  290, 

a   =  .000299, 

c  =  7025, 

b   =  704, 

c  =  356, 

c  =  .000180, 

b  =  8530, 

C   =  73°. 

C  =  41°  10'. 

A  =  63°  .50'. 

C  =  40°. 

37. 

38. 

39. 

40. 

b   =  1482, 

a  =   .2785, 

B  =  50°  20'  54", 

C=  49°  47' 26 

a  =   12S4, 

b  =  .2275, 

a  =  235.64, 

c  =  725.52, 

.4  =  27°  18'. 

B  =  65°  40'. 

b   =  284.31. 

b   =950.04. 

In  any  triangle  ABC,  whose  sides,  opposite  angles  A,  B,  C,  respectively, 
are  o,  b,  c,  show  that: 

41.  b{s-b)  cos2 1  =a{s-a)  cos2  ^  • 

42.  a  =  b  cos  C  +  c  cos  fi. 

43.  (a  -  6)  (1  +  cos  C)  =  c  (cos  B  -  cos  A). 
a2+  62  +  c2 


. .     cos  A    ,    cos  jS   ,  cos  C 

44. r 

a  6  c 


2o6c 


45.    (b  +c  -  a)  tan 


.4 


{c  +a  -b)  tan  -  ■ 


46.  {b  +  c)  (1  -  cos  A)  =  a  (cos  B  +  cos  C). 

47.  (a2  -  62  +  c=)  tan  B  =  (n=  +  &"  -  C")  tan  C. 


48.    cot  2  +  cot  7j  +  cot  2 


.^      .5      .^ 
cot  2  cot  2  ^ot  2  ■ 


49.    The  radius  of  the  inscribed  circk 


V— 


)(s-b)(s-  c) 


50.   The  diameter  of  the  circumscribed  circle  is  o  esc  A. 

Calculate  x  in  terms  of  the  other  quantities  in  each  figure  below,  where  a 
right  angle  is  indicated  by  a  double  arc;  in  each  case  find  the  value  of  x  for  an 
assumed  set  of  values  of  the  literal  quantities: 


158 


OBLIQUE  PLANE  TRIANGLES 


[179 


63.  Find  the  lengths  of  diagonals  and  the  area  of  a  parallelogram  two  of 
whose  sides  are  5  ft.  and  8  ft.,  their  included  angle  being  60°. 

64.  Two  sides  of  a  parallelogram  are  a  and  b,  their  included  angle  C;  show 
that  the  area  is  ab  sin  C. , 

65.  The  sides  of  a  triangle  are  4527,  7861,  6448;  find  the  length  of  the 
median  drawn  to  the  shortest  side. 

66.  The  sides  of  a  triangle  are  in  the  ratio  of  2  :  3  :  4;  find  the  cosine  of 
the  smallest  angle. 

67.  The  angles  of  a  triangle  are  as  3  :  4  :  5;  the  shortest  side  is  500  ft.; 
solve  the  triangle. 

68.  The  angles  of  a  triangle  are  as  1  :  2  :  3;  the  longest  side  is  100  ft.; 
solve  the  triangle. 

69.  From  a  station  on  level  ground  due  south  of  a  hill,  the  angle  of  eleva- 
tion of  the  top  is  15°;  from  a  point  2000  ft.  east  of  this  station  the  angle  of 
elevation  is  12°;  how  high  is  the  hill  ? 

70.  The  angle  of  elevation  of  the  top  of  a  building  100  ft.  high  is  60°;  what 
will  be  the  angle  at  double  the  distance  ?     . 

71.  A  flag-pole  on  a  building  subtends  an  angle  of  7°  40'  at  a  point  on  the 
ground  500  ft.  from  the  building;  on  approaching  100  ft.,  the  pole  subtends 
an  angle  of  7°  50' ;  find  the  height  of  the  pole  and  the  building. 

■  72.  On  level  ground,  250  ft.  from  the  foot  of  a  building,  the  angles  of  ele- 
vation of  the  top  and  bottom  of  a  flag-pole  surmounting  the  building  are 
38°  43'  and  31°  2'  respectively;  find  the  height  of  the  building  and  the  pole. 

73.  From  level  ground  the  angle  of  elevation  of  the  top  of  a  hill  is  11°  30'; 
after  approaching  3000  ft.  up  an  incline  of  3°  27',  the  angle  of  elevation  of  the 
top  is  21°  32';   how  high  is  the  hill  ? 

74.  From  a  level  plain,  the  angle  of  elevation  of  a  distant  mountain  top 
is  5°  50';  after  approaching  4  miles,  the  angle  is  8°  40';  how  high  is  the  moun- 
tain ? 


179]  OBLIQirE  PLANE  TRIANGLES  159 

75.  From  a  point  GO  ft.  above  sea  level  the  angle  between  a  distant  ship 
and  the  sea  horizon  (the  offing)  is  20';  how  far  away  is  the  ship  ?  [Consider 
the  surface  of  the  sea  as  a  plane,  and  the  distance  to  the  horizon  10  miles. 
See  (226)  ex.  (4).] 

76.  From  a  point  on  level  ground  the  angle  of  elevation  of  the  top  of  a  hill 
is  14°  12';  on  approaching  1000  ft.,  the  angle  is  17°  50';  how  high  is  the  hill  ? 

77.  A  building  surmounted  by  a  flag-pole  20  ft.  high  stands  on  level  ground. 
From  a  point  on  the  ground  the  angles  of  elevation  of  the  top  and  the  bottom 
of  the  pole  are  53°  5'  and  45°  11'  respectively.     How  high  is  the  building  ? 

78.  On  approaching  1  mile  toward  a  hill,  the  angle  of  elevation  of  its  top 
is  doubled;  on  approaching  another  mile,  the  angle  is  again  doubled;  how  high 
is  the  hill  ? 

79.  A  and  B  are  two  points  neither  of  which  is  visible  from  the  other.  To 
determine  the  distance  AB,  two  stations  C  and  D  are  chosen  and  the  following 
measurements  made:  CD=500ft.;  ZACD  =  30°25' 15";  Z  ACB=  S5°  iO' 20"; 
Z  BDC  =  35°  14'  50";  Z  BDA  =  80°  20'  25";  find  AB. 

80.  In  a  chain  of  three  non-overlapping  triangles,  the  following  data  are 
known : 

AB  =  1000  ft. 
A  ABC,  AACD,  ZCDE, 

■  Z  A  =  44°  36',  Z  A  =  56°  32',  Z  C  =  55°  30', 

ZC  =40°  0';  ZC  =50°20';  Z.&  =  77°02'; 

Calculate  DE.     (Express  DE  in  terms  of  AB  and  the  necessary  angles  by 
the  law  of  sines.) 

81.  In  a  chain  of  four  non-overlapping  triangles,  the  following  data  are 
known: 

AB  =  11289  meters. 

A  ABC,  ACBD,  ADBE,  ADEF, 

Z  A  =  58°  10'  35",  Z  B  =  86°  50'  0",  Z  D  =  79°  12'  8",     Z  D  =  50°  41'  5", 

Z  5  =  69°  55'  0";     Z  C  ==  46°  48'  0";  Z  fi  =  73°  29'  10";  Z  ^  =  45°  20'  40"; 

calculate  EF. 

82.  In  a  chain  of  five  consecutive  triangles,  each  having  a  side  in  common 
with  the  preceding,  as  ABC,  CBD,  BDE,  DEF,  EFG,  express  FG  in  terms 
of  AB  and  the  necessary  angles. 

83.  A  tower  50  ft.  high  stands  on  the  edge  of  a  cliff  150  ft.  high.  At  what 
distance  from  the  foot  of  the  cliff  will  the  tower  subtend  an  angle  of  5°  ? 

84.  The  sides  of  a  triangle  are  100,  150,  200  ft.  At  the  vertex  of  the 
smallest  angle  a  line  100  ft.  long  is  drawn  perpendicular  to  the  plane  of 
the  triangle.  Find  the  angles  subtended  at  the  farther  end  of  this  line  by 
the  sides  of  the  triangle. 

85.  A  right  triangle  whose  perimeter  i.s  100  ft.  rests  with  its  hypotenuse 
on  a  plane,  the  vertex  of  the  right  angle  being  10  ft.  from  the  plane.  The 
angle  between  the  plane  of  the  triangle  and  the  supporting  plane  is  30°.  Find 
the  sides  of  the  triangle. 


160  OBLIQUE  PLANE  TRIANGLES  [179 

86.  An  equilateral  triangle  50  ft.  on  a  side  rests  with  one  side  on  a  plane 
with  which  its  plane  makes  an  angle  of  60°.  How  far  is  the  third  vertex 
from  the  plane  ? 

87.  As  in  exercise  86,  if  the  triangle,  instead  of  being  equilateral,  has  sides 

40,  20,  30  ft.  and  rests  on  the  shortest  side.     Ans.   —^ 

88.  The  sides  of  a  triangle  are  as  1  :  2  :  3,  and  the  longest  median  is  10  ft. 
Find  the  sides  and  angles. 

89.  The  following  measurements  of  a  field  ABCD  are  made:  A  to  B,  due 
north,  10  chains;  B  to  C,  N  30°  E,  6  chains;  C  to  D,  due  cast,  8  chains;  cal- 
culate AD,  and  the  area  of  the  field  in  acres.     (1  chain  =  4  rods.) 

90.  The  following  measurements  of  a  field  ABODE  are  made:  A  to  B,  due 
east,  25.52  chains;  J5  to  C,  E  40°  26'  N,  22.25  chains;  C  to  A  N  48°  26'  W, 
33.75  chains;  DioE,W  31°  15'  S,  18.32  chains;  calculate  EA  and  the  area  of 
the  field  in  acres. 

91.  In  the  field  of  exercise  89  how  much  area  is  cut  off  by  a  line  due  east 
through  B  ? 

92.  In  the  field  of  exercise  90  where  should  an  east  and  west  line  be  drawn 
so  as  to  bisect  the  area  ? 

93.  In  the  field  of  exercise  90  where  should  a  north  and  south  line  be 
drawn  to  cut  off  30  acres  from  the  western  part  of  the  area  ? 

94.  If  P  be  the  pull  required  to  move  a  weight  Tf  up  a  plane  inclined  to 
the  horizontal  at  an  angle  i,  and  m  the  coefficient  of  friction,  then 

p  _  TT7  sin  i  +  M  cos  i 
cos  i  —  M  sin  i 

Calculate  P  when  W  =  1000  lbs.,  i  =  30°,  /x  =  0.1. 

95.  In  exercise  94,  what  is  i  if  P  =  J  T7  and  ^  =  0.1  ? 

96.  If  I  be  the  length  of  a  plane  inclined  to  the  horizontal  at  an  angle  i, 
fi  the  coefficient  of  friction  and  g  the  acceleration  due  to  gravity  (32.  +  ft. 
per  sec.  per  sec.)  the  time  in  seconds  required  by  a  body  to  slide  down  the 
plane  is  / ^-^ 

y  g  (sin  i  —  m  cos  i) 

What  is  T  when  I  =  25  ft.,  i  =  20°,  m  =  0.1  ? 

97.    In  exercise  96,  find  i  when  I  =  100  ft.,  m  =0.1,  T=5  sec. 
W  /  98.    When  light  passes  from  a  rarer  to  a  denser  medium,  the 

index  of  refraction  m  is  determined  by  the  equation 
__  •  _  sin  i 

:-~  '^  ~  sin  r 

^-l^l--""  When  M  =  1-2,  what  must  be  i  (angle  of  incidence)  to  give  a 
"'■'  deflection  of  10°  ? 

99.  Find  the  total  deflection  of  a  ray  which  passes  through  a  wedge  whose 
angle  is  30°  and  index  of  refraction  1.4,  if  the  ray  enters  the  wedge  so  that  the 
angle  of  incidence  is  25°,  and  moves  in  a  plane  ±  to  the  edge  of  the  wedge.  _ 

100.  Solve  exercise  99  when  the  angle  of  the  wedge  is  a,  the  angle  of  mci- 
dence  i,  and  the  index  of  refraction  m. 


CHAPTER   XI 

The  Progressions.     Interest  and  Annuities 

180.  Arithmetic  Progressions.  —  Let  a,h,c,  .  .  .  ,  k,  I  be  quan- 
tities such  that  the  difference  between  any  one  of  them  and  the 
preceding  one  is  constant.  Then  the  quantities  are  said  to  form 
an  arithmetic  progression.     (We  shall  abbreviate  this  into  A.  P.) 

The  quantities  a,  h,  c  .  .  .  ,k,  I  are  called  the  terms  of  the  pro- 
gression, a  and  I  the  extremes,  and  h,  c,  .  .  .  ,k  the  means.  The 
constant  difference  between  consecutive  terms  is  called  the 
common  difference. 

Let  a  denote  the  first  term, 
I  denote  the  last  term, 
d  denote  the  common  difference, 
n  denote  the  number  of  terms, 
S  denote  the  sum  of  the  terms  of  any  A.  P.     Then 
the  second  term  is   a  ^  d, 
the  third  term  is     a  -f  2  d, 

the  last  or  nth  term  is  a  -\-{n  —  1)  d;   that  is, 
(1)  I  =  a+  {n  -  1)  d. 

Also 


S  =  a+(a  +  f/)  +  (a  +  2d)+   •  •  •   +(a  +  n  -  If/); 
S  =  l  +{l  -d)-\-(l  -2f/)+  •••+(/  -7T=n[f/). 
Adding, 

2.S=(a  +  0  +  (a  +  0+   •  •  •   +(«  +  /)=  n(a  +  0. 
Hence 

(2)  S  =  r^{a  +  l). 

Putting  for  I  its  value  from  (1),     ■ 

161 


162  THE   PROGRESSIONS  [181,  182 

We  shall  refer  to  the  five  quantities  a,  I,  d,  n,  S,  as  the  elements 
of  the  A.  P.  When  any  three  elements  are  given,  the  other  two 
may  be  found  by  use  of  the  preceding  formulas. 

181.  Problem.  To  insert  m  arithmetic  means  between  two  given 
quantities,  a  and  I. 

Since  there  are  2  extremes  and  m  means,  the  total  number  of 
terms  is  m  +  2.     Hence  if  d  be  the  common  difference, 

l  =  a-\-{m-\-2-  l)d; 
hence 

J       I  —  a 

Then  the  required  means  are 

a  +  rf,  a  +  2d,  .  .  .  ,  a  +  md. 

When  w  =  1  we  have  only  a  single  mean,  called  the  arithmetic 
mean.     It  equals  ^  (a  +  0. 

182.  Examples. 

1.  Find  the  sum  of  all  the  integers  from  1  to  100  inclusive, 
Here  5  =  1+2+3+  •  •  •   +  100. 
Then  a  =  1,  Z  =  100,  n  =  100, 

and  S  =  ^  (a  +  0  =  I  X  100  (1  +  100)  =  5050. 

2.  How  many  terms  of  the  progression  3,  0,  -  3,  .  .  .  are  required  to 
make  the  sum  equal  —  27. 

Here  a  =  3,  d  =  -  3,  S  =  -  27;  to  find  n. 

From  (2'),  -27  =  n  U -""^  X3^,    or    n2  -  3  n  -  18  =  0. 

Hence  «  =  6     or     -  3. 

Since  n  must  be  positive  we  discard  the  second  value. 

3.  Find  four  numbers  in  A.P.,  such  that  the  sum  of  the  first  and  last  shall 
be  12  and  the  product  of  the  middle  two  32. 

Let  the  numbers  he  a  —  3  d,  a  —  d,  a  +  d,  and  a  +  3  d,  with  a  common 
difference  2  d. 

Then  a-2d+a  +  3d  =  12 

and  ^  (a-d)ia+d)  =  32. 

Hence  a  =  6     and     d  =  ±  2. 

Therefore  the  numbers  are 

0,  4,  8,  12,    or    12,  8,  4,  0. 


1S3,  184]  THE  PROGRESSIONS  163 

183.  Exercises.  Find  the  last  term  and  the  sum  of  each  of  the 
following  arithmetic  progressions: 

1.  7,  11,  15,  .  .  .  ,  to  13  terms;  5.  63,  58,  53,  .  .  .  ,  to  8  terms; 

2.  5,8,11.  .  .  .  ,  to  12  terms;  g^  x,x+2y,x+ i  y, .  .  .  ,  tolOterms; 

3.  2,  2i,  3,  .  .  .  ,  to  25  terms; 

4.  1,1.1,1.2,..  .  .  ,to200terms;  '^'  V,V-hq,V-q,.  ..,  to  20  terms. 

Find  the  other  elements  of  the  A. P.,  given  that : 

8.  a  =  10,  n  =  14,  S  =  1050;  16.   n  =  35,  S  =  2485,  d  =  3; 

9.  a  =  3,  n  =  50,  S  =  3825;  17.   n  =  50,  >S  =  425,  d  =  h 

10.  a  =  -  45,  n  =  31,  'S  =  0;  18.  n  =  33,  S  =  -  33,  ri  =  -  f; 

11.  Z  =  21,  n  =  7,  S  =  105;  19.  S  =  624,  a  =  9,  rZ  =  4; 

12.  /  =  49,  n  =  19,  5  =  503J;  20.  S  =  2877,  a  =  7,  d  =  3; 

13.  Z  =  148,  n  =27,  S  =  2241;  21.  S  =  623,  d  =  5,  I  =  77; 

14.  Z  =-143,n  =  33,  S=-2079;  22.  S  =  682.5,  rf  =  1.5,  Z  =  45; 

15.  n=  21,  S  =  1197,  d  =  4;  23.  S  =  95172,  d  =  -  7,  Z  =  567. 

24.  Find  the  sum  of  the  first  100  odd  numbers. 

25.  Find  the  sum  of  the  first  50  multiples  of  7. 

26.  A  body  starting  from  rest  falls  16  ft.  during  the  first  second,  and  in 
every  other  second  32  ft.  more  than  during  the  preceding.  How  far  does  the 
body  fall  in  12  seconds;  how  far  during  the  12th  second  ? 

27.  According  to  the  rate  of  fall  in  exercise  26,  how  long  will  the  body  take 
to  fall  1600  ft  ? 

28.  A  body  which  is  projected  vertically  upward  loses  32  ft.  of  its  initial 
velocity  each  second.  If  the  velocity  of  projection  is  320  ft.  per  second,  how 
high  will  the  body  rise  ? 

29.  If  100  apples  are  laid  in  a  straight- line,  3  feet  apart,  how  far  must  a 
person  walk  to  carry  them  one  at  a  time  to  a  basket  standing  beside  the  first 
apple  ? 

184.  Geometric  Progressions.  —  If  the  numbers  a,  b,  c,  .  .  .  , 
k,  I  are  such  that  the  ratio  of  any  number  to  the  preceding  number 
is  constant,  the  numbers  form  a  geometric  progression.  (We 
abbreviate  by  writing  G.  P.) 

The  expressions  "terms,"  ''means,"  "extremes,"  are  used  here  as 
in  the  case  of  A.  P.  The  constant  ratio  of  any  term  to  the  preced- 
ing is  called  the  ratio  of  the  geometric  progression. 

If  a,  I,  n,  and  S  have  the  same  meaning  as  in  the  case  of  the 
A.  P.,  and  if  r  denote  the  ratio  of  the  G.  P.,  the  first  n  terms  are, 
a,  ar,  ar^,  ar^,  .  .  .  ,  ar''-'^. 


164  THE   PROGRESSIONS  [185,  186 

Hence 

(1)  1  =  ar''-^. 

Also  S  =  a -{- ar -{- ar^ -\-   •  •  •   -\-ar"'-'^ 

and  rS  =  ar  -{-  ar-  -{-  •  •  •  +  ar"  -  ^  -|-  ar"*. 

Therefore  rS  —  S  =  ar"  —  a, 

or  (r-l)S=  (r"  -  1)  a. 

Hence 

■y"  —  1         1  —  *■"' 

(2)  S  =  a- -  =  a- 

Substituting  from  (1)  in  (2)  we  have 

(20  s  =  ':^. 

When  any  three  of  the  five  elements  are  given,  the  other  two 
may  be  obtained  by  use  of  two  of  the  preceding  formulas.  In 
some  cases  this  involves  the  solution  of  an  equation  of  nth  degree 
or  of  an  exponential  equation, 

185.  Problem.  To  insert  m  geometric  means  between  two  given 
numbers  a  and  I. 

The  total  number  of  terms  being  m  +  2,  we  have,  if  r  denote  the 

ratio, 

m+in 

I  =  ar'^  +  s-^     or     r  =  V  -. 
▼  a 

The  required  geometric  means  are  then 

ar,  ar^,  .  .  .  ,  ar^. 

When  m  =^  1,  the  resulting  single  mean  between  a  and  I  is 
y/al.  The  square  root  of  the  product  of  two  quantities  is  called 
their  geometric  mean. 

186.  Examples. 

1.  Find  the  sum  of  the  first  10  terms  of  the  G.  P.  2,  22,  23,  .  .  .     . 

210  —  1 
Here  a  =  2,    r  =  2,  w  =  10;  hence S  =2  =  2046. 

2.  How  many  terms  of  the  G.  P.  1,  2,  4,  .  .  .  are  required  to  make  the 
sum  63  ? 

Here  a  =  1,  r  =  2,  -S  =  63;  to  find  n. 

From  S  =  a  ^"  ~}     we  have    63  =  ^!!  ~  ]  ;    or,  64  =  2«. 

Hence  n  =  6. 


187,  188]  THE  PROGRESSIONS  165 

3.    Four  numbers  are  in  geometric  progression.     The  sum  of  the  first  and 
last  is  18,  the  product  of  the  second  and  third  32.     Find  the  numbers. 
Let  the  numbers  be  a,  ar,  ar~  and  ar^. 
Then 

(1)     a+ar3  =  18;         (2)     02,-3  =  32. 

Multiply  (1)  by  a  and  in  the  result  replace  a?r^  by  32. 

Then  a^  +  32  =  18  a;     hence     a  =  16  or  2. 

Substituting  the  values  of  a  in  (2)  we  find  r  =  !  or  2.     Hence  the  numbers  are 

16,  8,  4,  2;  or  2,  4,  8,  16. 
(We  disregard  the  imaginary  values  of  r.) 

187.  Exercises.  Find  the  last  term  and  the  sum  of  the  terms 
of  the  following  geometric  progressions: 

1.  4,8,16,  .  .  .  ,  to  7  terms.  4.    9,  3,  1,  .  .  .  ,   to  11  terms. 

2.  2,6,18,  .  .  .  ,  to  9  terms.  6.    1,  ',,  i',,  .  .  .  ,  to  10  terms. 

3.  1,  4,  16,  .  .  .  ,  to  7  terms.  6.    8,  2,  f ,  to  20  terms. 

7.  a,  a  (1  +  x), a  (1  +  xY,  ...  to 8  terms. 

8.  rrfi,mn,m.-hi^,  .  .  .  ,  to  9  terms. 

9.  Insert  3  geometric  means  between  8  and  10368. 

10.  Insert  5  geometric  means  between  2  and  31250. 

11.  Insert  5  geometric  means  between  36  and  /j. 

12.  Insert  6  geometric  means  between  3  and  49152. 

13.  Insert  4  geometric  means  between  48  and  o\- 

14.  Insert  5  geometric  means  between  81  and  V/- 

Calculate  the  unknown  elements,  given : 

15.  Z  =  128,  r  =  2,  n  =  7.  22.  a  =  \,         Z  =2401,  S  =  2801. 

16.  Z  =78125,  r  =  5,  n  =  8.  23.  a  =  10,       I  =  h,  5  =  191-^. 

17.  l=i^,  r  =  \,  n  =  5.  24.  a  =  3125,  I  =b,  5  =  3905. 

18.  a=9,  ^  =  2304,  r  =  2.  25.  a  =  3,        r  =3,  5  =  29523. 

19.  0=2,  Z  =  64,  r  =  2  26.  a=8,         r  =2,  5  =  4088. 

20.  rt  =  3,  /  =  192V2,  r=V2.  27.  r  =  2,         n  =  7,  5  =  635. 

21.  0=2,  Z  =  1458,  5  =  2186.  28.  Z  =  1296,  r  =6,  5  =  1555. 

188.  Infinite  Geometric  Progressions.  —  Consider  a  line  segment 
AB  of  unit  length,  and  bisect  it  at  Ai,  then  bisect  Ai5  at  A-z,  A2B 
at  .4.3  and  so  on  (figure). 

The  points  of  bisection  A,,  A.,  ^3,  •  •  •   ^ ^ ^''  '^^  ^^^ 

continually  approach  B  and  the  sum  of  --     -  - 

the  segments  AA\  4-^1^12  +  MA?,  +  •  •  •  approaches  AB  or  1. 

But  the  sum  of  these  segments  is  represented  numericallj^  by  the 

series 

^  +  i  +  §+'  •  •'     ^^     2  +  2^  +  23+'  •  •' 


166  THE  PROGRESSIONS  [189 

and  hence  by  taking  n  large  enough  we  can  make  the  sum 

^    =14-1-4-14-   .    .    .   -^1 
^n      2  ^  22  "^  23  "•"  "^  2" 

differ  from  1  by  as  little  as  we  please.     Hence  we  take 
2  +  4  +  8+  ■  ■  ■  toii^finity  =  1. 

The  sum  Sn  above  is  a  geometric  progression  with  r  =  I  and 
a  =  |.     Its  sum  to  n  terms  is  therefore 

1  (hr  - 1 

As  n  increases,  {\Y  approaches  0,  and  Sn  approaches  the  value 

-  y =  1,  as  found  above. 

2  2  —  1 

A  geometric  progression  in  which  the  number  of  terms  increases 
without  limit  is  called  an  infinite  geometric  progression. 

For  the  sum  of  n  terms  of  any  G.  P.  we  have 
r'^  -  \  1  -  r" 

If  now  r  <  1,  then  r'^  approaches  0  when  n  approaches  oo,  and  the 
formula  for  the  sum  of  an  infinite  G.  P.  is 

S  =     _    >     provided  |  r  |  <  1. 

(When  r  =  1,  or  when  r  >  1,  -S  is  infinite.) 

Exam-pie.  A  ball  is  thrown  vertically  upward  to  a  height  of  60  ft.  On 
striking  the  ground  it  always  rebounds  to  one-third  the  height  from  which 
it  fell.     How  far  will  it  travel  ? 

The  distance  covered  during  the  first  rise  and  fall  is  120  ft.,  during  the  sec- 
ond rise  and  fall,  h  X  120  ft.,  during  the  third,  ^^  X  120  ft.,  and  so  on  indefi- 
nitely. We  have  an  infinite  G.  P.,  with  a  =  120  and  r  =  J.  Hence  the  total 
distance  will  be 

S  =  j^  =  180  ft. 

189.  Exercises.  Sum  the  following  infinite  geometric  progres- 
sions: 

1.  8,  2,  *,  .  .  .    .         3.  5,  3,  § 5.   1,  -h,  +i  -i,    .... 

2.  \,\,-h,  ....  4.  2,?,  A,  ...    .        6.   3,  -1,  i  -I,    .  .  .    . 


190,  191  ]  THE  PROGRESSIONS  167 

7.  If  in  the  example  worked  above  the  ball  requires  4  seconds  for  the  first 
rise  and  fall,  and  half  as  much  time  for  any  subsequent  rise  and  fall  as  for  the 
preceding,  how  long  before  the  ball  will  come  to  rest  ? 

8.  How  far  has  the  ball  in  the  above  example  traveled  at  the  10th  rebound  ? 

190.  Harmonic  Progressions.  —  If  the  numbers  a,b,  c,  .  .  .  ,  k, 
I  are  such  that  their  reciprocals  form  an  arithmetic  progression, 
they  are  said  to  be  in  harmonic  progression  (abbreviated  to  H.  P.). 

Problems  relating  to  harmonic  progressions  are  solved  by  reduc- 
tion to  A.  P. 

If  a,  b,  c  form  a  H.  P.,  then  b  is  called  the  harmonic  mean  between 
a  and  c.     Let  the  student  show  that  we  then  have 

,        2  ac 
b  =  — , — 

191.  Exercises. 

1.  In  an  A.  P.  the  sum  of  the  9th  and  12th  terms  is  40;  the  difference 
between  the  squares  of  the  15th  and  11th  terms  is  400.     Find  a  and  d. 

2.  In  an  A.  P.  of  10  terms,  the  sum  of  the  terms  is  65  and  the  sum  of  their 
squares  1 165.     Find  a  and  d. 

3.  In  an  A.  P.  of  20  terms,  the  sum  of  the  3rd  and  12th  terms  is  30,  the 
product  of  the  two  middle  terms  is  725.     Find  a  and  d. 

4.  In  an  A.  P.  of  14  terms,  the  product  of  the  first  and  the  last  is  276  and 
the  product  of  the  middle  two  is  1326.     Find  a  and  d. 

5.  Find  four  numbers  in  A.  P.  such  that  their  product  is  840  and  their 
sum  11. 

6.  Find  four  numbers  in  A.  P.  such  that  their  product  is  h  and  the  sum  of 
their  squares  is  k. 

7.  Find  five  numbers  in  A.  P.  such  that  their  product  is  a,  their  sum  5  6. 

8.  The  sides  of  a  triangle  form  an  A.  P.  with  a  common  difference  2.   Find 
the  cosine  of  the  largest  angle,  if  the  longest  side  is  twice  the  shortest. 

9.  Find  the  angles  of  a  triangle  if  they  form  an  A.  P.  with  d  =  5°. 

10.  Between  every  pair  of  consecutive  terms  of  the  G.  P.  1,  2,  4,  8,  .  .  . 
insert  a  new  term  so  that  the  result  is  again  a  G.  P. 

11.  As  in  exercise  10  for  the  G.  P.  a,  ar,  nr^,  .... 

12.  In  a  G.  P.  of  10  terms,  the  sum  of  the  even  terms  is  30  and  of  the  odd 
terms  60.     Find  a  and  r. 

13.  Find  four  numbers  in  G.  P.  such  that  the  product  of  the  first  and  last 
is  400  and  the  quotient  of  the  middle  two  is  14. 

14.  Find  three  numbers  in  G.  P.  such  that  their  sum  is  h,  the  sum  of  their 
squares  k. 

15.  If  a  tree,  now  4  inches  in  diameter,  increases  its  diameter  5%  each 
year,  how  thick  will  it  be  in  20  years  ? 

16.  A  seed  yields  a  plant  from  which  4  new  seeds  are  obtained.  How  many 
seeds  are  available  from  the  10th  generation  of  plants  ? 


168  INTEREST  AND  ANNUITIES  [192 

17.  An  Indian  potentate  offered  to  reward  the  inventor  of  the  game  of  chess 
as  follows :  one  grain  of  wheat  for  the  first  square  on  the  chessboard,  2  for  the 
second,  4  for  the  third,  and  so  on,  doubling  each  time  for  the  64  squares.  What 
would  be  the  cash  value  of  this  reward,  with  wheat  at  $1.00  a  bushel,  allow- 
ing a  million  grains  to  the  bushel  ? 

18.  A  right  triangle  has  a  hypotenuse  2  ft.,  angle  30°.  From  the  vertex 
of  the  right  angle  a  _L  is  dropped  on  the  hypotenuse,  forming  a  new  right 
triangle  which  is  treated  similarly,  and  so  on  indefinitely.  Find  the  sum  of 
all  the  Js  so  obtainable. 

19.  The  altitude  of  an  equilateral  triangle  is  a.  A  circle  is  inscribed  in  it, 
and  in  this  circle  a  new  equilateral  triangle.  The  operation  is  repeated  on  the 
new  triangle,  and  so  on  indefinitely.  Find  the  sum  of  the  altitudes  and  of 
the  perimeters  of  all  triangles  so  obtainable. 

20.  Find  the  sum  of  the  perimeters  and  of  the  areas  of  all  the  circles  in 
exercise  19. 


Interest  and  Annuities.  —  This  subject  affords  a  simple  and  use- 
ful application  of  the  theory  of  progressions. 

192.  Interest.  —  Let  P  denote  a  sum  of  money  loaned,  or 
principal,  and  r  the  yearly  rate  of  interest  expressed  in  fractions 
of  a  dollar.     Then  the  amount  of  P  dollars  in  one  year  is 

A,  =P(l+r). 

If  principal  plus  interest  for  one  year  is  allowed  to  run  a  second 
year,  the  amount  at  the  end  of  the  second  year  is 

A2  =  Ai(l-\-r)=P(l+rr, 
and  so  on. 

Hence  ii  Anhe  the  amount  of  P  dollars  in  n  years,  interest  at 
rate  r  compounded  annually,  we  have 

(1)  A,,  =  P(l-\-rr. 

If  interest  is  compounded  every  t  years  instead  of  annually,  then, 
after  n  compoundings,  the  amount  is 

(!')  _  ^„  =  P  (1  -f  rty\ 

Thus  if  we  want  the  amount  of  $100  at  the  end  of  2  years,  inter- 
est 4  per  cent  compounded  quarterly,  we  have, 

P  =  $100;  r  =  t!o;  t  =  \;  n  =  8. 

Then     An  =  100  (1  -f  .04  X  i)^  =  $100  (1.01)^  =  $108.25. 


193]  INTEREST  AND  ANNUITIES  169 

193.  Annuities.  —  An  annuity  is  a  sum  of  money  payable  yearly, 
or  at  other  stated  periods. 

Let  A  be  the  amount  of  each  payment,  r  the  yearly  rate  of 
interest,  n  the  number  of  payments  to  be  made. 

Assuming  the  first  payment  now  due,  and  that  each  payment  is 
put  at  interest,  compounded  annually,  what  is  the  total  amount 
accrued  when  the  last  payment  has  been  made? 

The  first  payment  is  at  interest  n  —  1  years,  its  amount 
A  (1  -{-  r)"-i;  the  second  n  —  2  years,  its  amount  ^  (1  +  r)"-^; 
and  so  on,  to  the  payment  next  before  the  last,  which  is  at  inter- 
est one  year,  its  amount  A  {1  -\-  r);  the  last  payment  amounts 
to  A.     The  total  amount  S  is  therefore 

AS  =  A+A(14-*-)+A(l+r)2+  •  •  •  +^(i-f-,.)n-i^  or 

(2^         ^-^^  1  +  ..-1  -^ — ; 

Present  Worth.  —  How  much  cash  in  hand,  placed  at  interest 
compounded  annually,  will  amount  to  the  sum  S  just  obtained 
when  the  last  payment  is  made,  that  is,  in  ?i  —  1  years? 

Let  Q  be  the  amount  required,  called  the  present  worth  of  the 
annuity. 

Let  Qi  be  the  sum  which  with  interest  will  yield  in  n  —  1  years 
the  amount  of  the  first  payment,  or  A  (1  +  r)"-^     Then 

Qi(l+r)«-i  =^(l+r)"-i     or     Q^  =  A. 

Let  Q2  be  the  sum  which  with  interest  for  w  —  1  years  will  yield 
the  amount  of  the  second  payment,  or  A  (1  -f  r)"-^.     Then 

Q2(l+r)"-i  =^(l+r)"-2     or     Q2  =      ^ 


1+r 


Similarly  if  Qs,  Q4,  .  .  .  Qn  be  the  present  worths  of'  the  3rd, 
4th,  .  .  .  last  payments  of  the  annuity  we  have, 


Hence 


170  INTEREST  AND  ANNUITIES  [194 

The  sum  in  the  parentheses  is  a  G.  P.  with  ratio  ^         •    Apply- 
ing the  formula  and  reducing, 

(1  +  rY  -  1 


(3)  Q 


194.  Exercises. 

1.  Find  the  amount  of  $1412  in  19  years  at  4%,  interest  compounded 
annually. 

2.  Find  the  present  worth  of  an  annuity  of  $100,  there  being  20  annual 
payments  of  which  the  first  is  now  due. 

3.  Find  the  amount  of  $1000  in  10  years  at  4%,  interest  compounded 
quarterly. 

4.  Find  the  amount  of  $1000  in  20  years  at  4%,  interest  compounded 
semi-annually. 

5.  In  how  many  years  will  a  sum  of  money  double  itself  at  5%  simple 
interest  ? 

6.  In  how  many  years  will  a  sum  of  money  double  itself  at  5%,  interest 
compounded  annually  ? 

7.  An  annuity  of  $100  is  to  begin  in  10  years  from  date  and  to  run  10 
years.     Find  its  present  worth  if  money  brings  5%  compound  interest. 

8.  Find  the  present  worth  of  a  perpetual  annuity  of  ^1  dollars,  compound 
interest  r%,  the  first  payment  now  due.     (Q  =  Qi  +  Q2  +  Qs  +  •  •  •  ad  inf.). 

9.  As  in  exercise  S,  except  that  the  first  payment  falls  due  in  m  years. 


CHAPTER  XII 

Infinite  Series 

195.  Limit  of  a  Variable  Quantity.  —  When  a  variable  quantity 
changes  in  such  a  way  that  it  approaches  a  fixed  numerical  value,  so 
that  the  difference  between  the  variable  and  the  fixed  quantity  becomes 
and  remains  less  than  any  assignable  magnitude,  however  small,  then 
the  fixed  quantity  is  called  the  limit  of  the  variable. 

For  example,  as  x  varies  the  variable  quantity  1  -{-  x  can  be 
made  to  differ  from  1  by  less  than  any  small  quantity  e,  by  simply 
taking  |  a:  |  <  e,  and  the  nearer  x  is  to  0,  the  nearer  will  1  -f  a;  be 
to  1.  Hence,  as  x  approaches  0,  the  limit  of  1  -\-  x  is  1.  As  an 
equation  this  is  expressed  by 

lim  (1  -|-  a;)  =  1.     (=  is  read  "approaches.") 

1  =  0 

Exercise .     Show  that : 
(a)  Hm  ^  =  1 ;     (6)  Hm  (l  +  J)  =  2;     (c)  lim  log  (1  +  a:)  =  0; 

(rf)  lim  fl  -  ^V  0;     (e)lime"=l;     (/)  lim  fl  + -Y  =  1. 

n  =  10\  n/  x^O  n  =  oo\  11/ 

196.  Infinite  Series.  — A  sequence  or  succession  of  terms,  ui,  U2, 
Us,  .  .  .  ,  iin,  .  .  .  ,  unlimited  in  number,  is  called  an  infinite  series. 

The  sum  of  the  first  n  terms  of  a  sequence  we  denote  by  5„. 
Then 

Sn  =  Ui  +  U2  +  U3  -\-     •    •    •     +  Un. 

As  n  increases  and  we  form  the  sum  of  more  and  more  terms  of 
the  sequence,  one  of  three  alternatives  is  open  to  5„,  namely: 

(a)  Sn  approaches  a  fixed  limit  S,  which  is  then  called  the  sum 
of  the  infinite  series,  and  the  series  is  said  to  converge. 

(b)  Sn  increases  without  limit;  the  infinite  series  then  has  no 
sum  and  is  said  to  diverge. 

(c)  Sn  oscillates;  the  infinite  series  has  no  sum  but  oscillates, 
and  is  again  said  to  diverge. 

171 


172  INFINITE  SERIES  [197 

Examples. 


(a)  2  "^  22  "^  23  "^  '  '  ■   "^  2"  "*"  ' 

1.1,        .11  i^r  - 1 


g.  (188. 


^^n      2  "^  22  "^  ■  ■  ■  "^  2''      2    i  -  1 
lim  *S„  =  1  =  >S.     The  series  converges  to  the  value  1, 

71  —  00 

or,  ^  +  ^2+  •  •  •  +^+  •••=!.   [(188),  figure.] 

(b)  l  +  2  +  3+---+n+.-.. 

*S„  =1  +  2  +  3+  •  •  •  -\-n;  then  obviously  Sn  increases  with- 
out limit  as  more  and  more  terms  are  added.  Hence  the  given 
series  has  no  sum,  and  diverges. 

(c)  1  -  I  +  1  -  1  +  •  •  •  •. 

Here  Si  =  1;  >S2  =  1  -  1  =  0;  ^Ss  =  1  -  1  +  1  =  1;  >S4  =  0;  and 
so  on  indefinitely.  Sn  oscillates  from  0  to  1  as  n  varies,  the  series 
is  oscillatory  and  has  no  sum.     We  say  that  it  diverges. 

197.  To  show  that  an  infinite  series  converges,  it  must  be  shown 
that  Sn,  the  sum  of  its  first  n  terms,  approaches  a  definite  limit  as  n 
increases  indefinitely.  When  such  limit  does  not  exist,  the  series  is 
divergent. 

The  direct  method  of  determining  whether  a  given  series  con- 
verges or  diverges  is  to  form  the  sum  of  its  first  n  terms  *S„,  and  let 
h  increase  indefinitely.  This  method  is  applicable  only  in  the 
few  cases  where  a  formula  for  Sn  is  available.  The  standard  case 
is  that  of  the  infinite  geometric  progression, 

a  -{-  ar  -\-  ar^  +   •  •  •  +  ar"'-^  +  •  •  • 

1—  r'* 

Here  *S„  =  a  +  ar  +  ar~  +   •  •  •  -\-  ar"^-^  =  a 


1-r 


When  r  is  numerically  less  than  1,  i.e.,  \r\  <  1,  then  r"  approaches 
0  as  n  increases  and 

lim  Sn  =  a     _     =  S. 
n  =  oo  -i        r 

When  r  =  1, 

Sn  =  a  -\-  a  +  •  •  '  -\-  a  =  na. 


198,  199]  INFINITE  SERIES  173 

Hence  Sn  increases  without  limit  when  n  increases.  When 
I  r  I  >  1,  r"  increases  indefinitely  with  n;  hence  S^  does  the  same. 
Therefore,  the  geometric  series,  a -\r  ar  -{-  «/■'+  •  •  •  >  converges  when 
\r\  <  1,  and  diverges  when  \r\  =1. 

Putting  a  =  1 ,  we  see  that  the  smpZe  pother  series,  I+cc+cc^tI-  •  •  •  ^ 
converges  when  \oc\  <  1  and  diverges  when  |  a?  |  =  1 . 

198.  We  next  consider  indirect  methods  for  establishing  the 
convergence  or  divergence  of  a  given  infinite  series. 

Theorem  1.  When  an  infinite  series  converges,  its  nth  term  ap- 
proaches zero  as  a  limit  when  n  increases. 

Proof.  Let  the  convergent  series  be  W1+W2+ Ms  +  •  •  •  +Wu+  •  •  • . 
Then      Sn  =  ui-{-U2-\-  •  •  •  +m„    and    Sn-i  =  ui-^U2-\-  •  •  •  +Wn-i. 

Hence  Un  =  Sn  —  Sn-i- 

By  taking  n  large  enough,  both  *S„  and  Sn-i  can  be  made  to 
differ  from  the  sum  of  the  series  and  hence  from  each  other  by 
as  little  as  we  please;  hence  their  difference,  w„,  can  be  made  to 
differ  from  zero  by  less  than  any  assignable  small  quantity. 

lim  Un  =  0. 

n  =  oo 

This  is  a  necessary  condition  for  the  convergence  of  any  series. 
Test  for  Divergence.  —  From  Theorem  1  we  infer  that  an  infinite 
series  diverges  whenever  lim  iin  9^  0. 

n  =  oo 

199.  Alternating  Series.  —  A  series  whose  terms  are  alternately 
+  and  —  is  called  an  alternating  series. 

Theorem  2.  An  alternating  series  converges  provided  that  (a) 
each  term  is  numerically  less  than  the  preceding,  and  (b)  the  linvit 
of  the  nth  term  is  zero  as  n  increases  indefinitely. 

Proof.     Let  the  series  be 

Wi  —  M2  +  Ms  —  M4  +  W5  —  Me  +   •  •  •  . 
Write  this  in  the  two  forms, 

(mi  -  M2)  +  {Uz  -  U4)  +  (M5  -  Uq)  +    •    •    •  ; 
Ml  -  (M2  -  M3)  -  (M4  -  Ms)  -     •    •    •    . 

Each  set  of  parentheses  incloses  a  positive  quantity  according  to 
condition  (a)  of  the  theorem;  hence  assuming  that  mi,  U2,  M3,  .  .  . 
are  themselves  positive  quantities,  the  first  form  shows  that  the 


174  ■     INFINITE   SERIES  [200,201 

sum  of  the  series  is  positive,  i.e.,  >  0,  and  the  second  that  the 
sum  is  less  than  the  first  term  wi.     Also,  since  hm  w„  =  0,  the  sum 

n  =  00 

cannot  oscillate.     Hence  the  series  converges  to  a  value  between 
0  and  its  first  term. 

Exam-pie.     The  alternating  series, 

l-h  +  \-\+  '  •  • 

converges  to  a  value  between  0  and  1. 

200.  Absolute  Convergence.  —  A  series  is  said  to  converge  abso- 
lutely when  it  remains  convergent  if  all  its  terms  are  taken  positively. 

Thus  if  wi,  M2,  W3,  •  •  •  be  in  part  negative  and  in  part  positive, 
the  series 

Wi  +  W2  +  W3  +    •    •    • 

converges  absolutely  provided  that  the  series 

I  Wi  1+  I  W2  I  +  I  W3  I  +    •    •    • 

converges. 

Exercise.     Show  that  the  series 

\  ■\- X -\- x"^  -{-  •  •  •     and     a -\- ax -^  ax^  +  -  •  • 

both  converge  absolutely  when  \x\  <  1. 

201.  The  Comparison  Test. 

Let  wi  +  W2  +  W3  +  •  •  • 

be  a  series  known  to  converge  absolutely  or  to  diverge. 
.      Let  vi -\- V2 -\- v^ -{-  ■  •  • 

be  a  series  to  be  tested  for  convergence  or  divergence.     Then, 

(a)  If  the  u-series  converges  absolutely  and,  for  all  values  of  n,  v^  is 
numerically  less  than  Un,  the  v-series  also  converges  absolutely; 

(b)  If  the  u-series  diverges  and  Vn  is  numerically  greater  than  Un, 
and  if  all  the  terms  of  the  v-series  have  the  same  sign,  the  v-series  also 
diverges. 

Proof. 

Let  r/n  =   I  Wl  I   +  I  W2  I   +   I  W3  I  +     •     •     •     +  \Un\ 

and  Vr,=  \vx\  +  \v2\  +  \vz\+  •  •  •   +|t'„|. 

Then  by  condition  (a),  Un  approaches  a  limit,  say  C/,  as  n  =  oo , 
and  also,  7„  <  f/„.     Hence,  since  F„  must  increase  steadily  with 


standard  Test  Series. 

(For  use  i: 

(1)   a  -\-  ax  -{■  ax^  +  • 

•  .  +  aa:'^  + 

(2)    l-\-x^x'+  .  ■ 

■+x'^+  ■  • 

(3)    l+^  +  |l+-  • 

•.^-H.. 

(4)    1  +  1+1+..  . 

+  ^--- 

(5)  j-^  +  ^^-^h^' 

■-^- 

201]  INFINITE   SERIES  175 

n,  but  is  always  less  than  f/„,  it  must  approach  a  limit  7,  less  than 
U.     Hence  the  v-series  converges. 

Under  condition  (b),  [/„  increases  without  limit,  and  also, 
Vn  >  Un.  Hence  F„  also  increases  without  limit  and  the  t^-series 
diverges. 


,  i  Conv.  when  |  a;  |  <  1 ; 
)  Div.  when  \  x  \  =1. 

Convergent. 

Divergent. 

^  Conv.  whenp  >  1; 
S  Div.  when  p  =  1. 

The  first  three  of  these  series  are  geometric  progressions  and  have 
already  been  considered. 

Series  (4)  can  be  shown  to  diverge  by  grouping  its  terms  thus : 

i+i  +  a  +  i)+(^+i  +  ^  +  ^)+(^+i^0+  •  •  •  +tV)+  •  •  • . 

We  can  form  in  this  way  an  infinite  number  of  parentheses,  each  of 
which  is  >  ^.     Hence  the  sum  is  infinite. 

Series  (5)  is,  term  for  term,  greater  than  or  equal  to  (4),  when 
p  =  1 ;  hence  for  these  values  of  p  the  series  diverges,  by  condition 
(b)  above.  When  p  >  1,  the  series  is  shown  to  converge  by 
grouping  its  terms  as  follows: 

p  +  (^2^  +  3]^  j  +  (^4^  +  •  •  *  +  7^  j  +  (s^  +  ■  •  ■  +  i5^j  +  •  •  •  . 

Considering  each  group  of  terms  as  a  single  quantity,  we  see  that 
this  series  is  less,  term  for  term,  than  the  series 

1+2+1+8 

2''      4^      8^ 

1.1.1,1. 

or  1  +  2?n  +  4^rri  +  g^i  +  •  •  •  • 

I 
2P- 

fore  the  given  series  converges. 


176  INFINITE  SERIES  [202 

Examples. 

1.  The  series  1+22  +  33+'  ■■+^+'''  converges;  for  it  is  less, 
term  for  term,  than  (3). 

2.  The  series  1  +^ ?,  +-, ^  +  •  •  •  + ,  „    ^  +  .  .  .  diverges;    for 

logio2       logio3  logion 

it  is  greater,  term  for  term,  than  (4). 

202.  The  Ratio  Test. —  The  series  Wl+M2+^*3+  ■  •  •  +Mra+  •  •  • 
converges  absolutely  if,  beginning  at  some  point  in  the  series,  the 
ratio  Uji  -r-  Un-i  becomes  and  remains  numerically  less  than  a  fixed 
positive  number  which  is  itself  less  than  1. 

Proof.     Assume  that 

^^    <  r  <  1  for  all  values  of  n  >  iV, 

\Un-l  I 

A^  being  a  fixed  positive  integer. 

Then  |  m„  |  <  r\  Un-\  \  when  n>  N. 

Hence  putting  n  =  A^  +  1,  A^  +  2,  .  .  .  ,  we  have 

\uN+i\<r\uN\; 

I  un+2  \  <  r\  un+1  I  <  r^\uN\; 

\uN+3\  <  r\uN+2\  <r^\uN\; 

Adding,  we  have 

I  UN+l  I  +  I  UN  +  2  \-\-UN+3\-\-  ■    ■    •  <  I  WiV  |(r  +  r2  +  r3  +  .    .    .). 
Writing  the  given  series  in  two  parts, 

(Wl  +  W2  +  •    •    •   +  Wiv)  +  (UN  +  1  +  Un  +  2  +  Un  +  3  +•••), 

we  see  that  the  first  part,  formed  of  N  terms  where  A^  is  a  fixed 
finite  integer,  must  have  a  finite  sum.  The  second  part  cannot 
exceed  the  left  member  of  the  last  inequality  above,  hence  is  less 
than  the  right  member  of  that  inequality.  But  the  series  r -{- r-  -{- 
7-3  _|_  .  .  .  converges  and  has  a  finite  sum,  since  it  is  a  G.  P.  with 
ratio  r  <  1.  Hence  the  sum  in  the  second  pair  of  parentheses  has 
a  finite  limit,  and  the  given  series  converges. 

Similarly  it  can  be  shown  that  the  series  diverges  when  the  test- 
ratio  Un^  Un-i  becomes  and  remains  greater  than  1,  or  even 
when  it  approaches  1  from  the  upper  side. 


203]  INFINITE  SERIES  177 

When  the  test-ratio  w„  -i-  w„_i  is  at  first  less  than  1,  but 
approaches  1  as  n  increases,  this  method  gives  no  information 
about  the  series. 

Examples. 

^•^+r^+TT^+--+1.2-3'....n+---- 

I    tf      I       1 
Here    — —    =  - ,  which  approaches  0  as  Ji  =  go  .     Hence  the  ratio  test 
\Un-l\        n' 

is  satisfied  and  the  series  converges. 

2.    sin  X  +  2  sin2  x  +  3  sin^  x  +  •  •  •   +  n  sin"  x  +  •  •  • 

I  'un    I  _  I  n  sin"  x         ' 

I  Wn -1  I       \  (n  —  l)  sin"- 1  x 

As  n  =  oo, =1,  and  if  we  choose  x  different  from  an  odd  multiple  of 

n  —  I 

^,  so  that  I  sin  X  I  <  1,  we  can  take  n  so  large  that  the  test-ratio  will  be 

less  than  r,  where  r  is  less  than  1.     We  need  only  take  x  <  sin-i  r— 

Ji 

Hence  the  series  converges  for  any  value  of  x  which  is  not  a  multiple  of  ^  • 


3. 

>+M 

+  ' 

.. .,!  +  ... 

Un 
Un- 

n  -  1 
1             n 

,  which  approaches  1 

from  the  lower  s 

4. 

12      3 

2  +  3+4 

+   • 

•■+nTl+- 

Un       _      n 
Un-l         n  +  1 

.71-1 

n 

n2  -  1 

Here  the  test-ratio  is  greater  than  1,  approaching  1  from  the  upper  side 
as  n  =  00 .  Hence  the  series  diverges.  This  series  may  also  be  shown  to 
diverge  by  comparison  with  (4)  of  (201). 

203.  Exercises.     Test  the  following  series: 

r        T-2  T?i 

3.  1  +  2  X  -I-  3  x2  4-  •  •  •   +  (n  +  1)  X"  -I-  •  •  •  . 

4.  cos  X  -f  cos2  X  4-  •  •  •   4-  cos"  x  +  •  •  •  . 

5.  tan  X  4-  tan2  x  4-  •  •  •   4-  tan"  x  4-  •  •  •  • 

6.  sin-ix  4- (sin-ix)2  4-  •  •  •   4- (sin-i  x)"  4-  •  •  •  . 

7.  logio  X  4-  (logio  x)2  4-  •  •  •  4-  (logio  x)"  4-  •  •  •  . 
„  1  -2    ,  2-2   ,  3-4 


178  INFINITE  SERIES  [203 

9    J-+A_  .  J_+.  .  . 
l-2^2-3^3-4^ 

10.*  \lx+\2x'i  +  \3x'+---+\nx^+---* 
11.    H-X  +  I+I+---. 

^2-    ^-IS  +  IS-JT+'-'- 
^.     -.       1       ,1-3          1-3.5,,     1.3-5-7, 
1*-    l-2^+2T4^-2-Tr:6^   +  2.4.6.8^ ' 

15.     Z   -  i  l2  +  4  X3   -  1  X-l  +    •    •    •    . 

*  [n  =  1  . 2  . 3  •  •  •  •  »!• 


CHAPTER  XIII 

Functions.     Derivatives.     Maclaurin's  Series 

204.  Functions.  —  Let  x  denote  a  variable  quantity  and  y  a 
quantity  whose  value  depends  on  that  of  x.  Then  y  is  said  to  be 
a  function  of  x.     Thus 

y  =  X-  -{-1,        y  =  a^,        y  =  sin  {ax  -\-  h) 

are  all  functions  of  x. 

As  an  equation,  we  indicate  that  y  is  a  function  of  x  by  writing 

y  =  f(x). 

When  a  body  is  dropped  from  rest,  the  space  s  (ft.)  fallen 
through  in  the  time  t  (seconds)  is  s  =  |  gf.  Here  s  is  a  function 
of  t,  or 

s=fit);       f(t)^hgt'- 

When  a  train  is  running  at  30  miles  an  hour,  the  space  s  (miles) 
covered  in  the  time  t  (hours)  is  s  =  30  t.     Hence 

s=f(t);        f(t)=^Qt. 

When  the  relation  between  y  and  x  is  given  by  an  equation  of 
the  form  y  =  f(x),  y  is  called  an  explicit  function  of  x. 

Suppose  the  relation  between  x  and  y  to  be  given  in  the  form, 

a;2 +  ?/  =  !. 

Here  y  is  not  given  directly  in  terms  of  x,  but  nevertheless  the 
value  of  y  depends  on  that  of  x;  for  when  we  substitute  for  x  first 
one  value  and  then  another  we  get  in  general  different  values  of  y 
on  solving  the  equation.  In  such  case  y  is  called  an  implicit 
function  of  x. 

As  other  examples,  we  have 

1/2  =  4  x"         sin  (x  -\-  y)=  1;        a""  -{-  a'-'  =  b. 

205.  Variation  of  Functions.  —  Consider  the  relation  y  =  x^. 
When  x  =  a,  then  y  =  a-;  when  x  =  a -\- h,  y  ={,a-\-  h)~. 

179 


180  DERIVATIVES  [206 

As  X  changes  from  a  to  a  -{-  h,  y  changes  from  a^  to  (a  +  h)-. 
The  total  change  in  x  is  h,  and  the  corresponding  change  in  y  is 
(a  +  hy  -  a2  or  2  ah  +  h^. 

Let  us  designate  a  change  in  x  by  Aa;  (read  "  increment  of  x," 
or  "delta  x")  so  that  in  this  example  Ax  =  h;  let  the  corre- 
sponding change  in  y  be  Ay,  so  that  we  have  in  this  case 

Ay  =  2ah-\-h^  =  2aAx  +  Ax^. 

In  general,  if  y  =  f  (x),  then  to  the  values  x  and  a:  +  Aa;  of  the 
variable  x  correspond  the  values  /  (x)  and  f{x-{-  Ax)*  of  y.  Hence 
the  change  in  y,  corresponding  to  the  change  Ax  in  x,  is 

Ay=f{x-\-Ax)-f{x). 

Continuous  Function.  —  When  Ay  =  0  with  Ax,  y  is  called  a 
continuous  function  of  x.  We  assume  all  our  functions  to  be 
continuous  unless  the  contrary  is  stated. 

Exercises. 

1.  Given  y  =  x^.     Calculate  Ay^  when  a;  =  2  and  Ax  =  0.1. 

2.  As  in  exercise  1,  when  y  =  's/x. 

3.  As  in  exercise  1,  when  y  =  x^. 

4.  As  in  exercise  1,  when  y  =  10-^. 

5.  Given  y  =  sin  x.     Calculate  Ay,  when  x  -  45°  and  Ax  =  5°. 

6.  As  in  5,  when  x  =  30°  and  Ax  =  1°. 

7.  As  in  5,  when  x  =  \  and  Ax  =  0.01. 

206.   Difference  Quotient.  —  The  fraction 

change  in  y  Av 

change  in  ..c  Ax 

is  called  the  difference  quotient  of  y  relative  to  x.  

Thus,  if  y  =  x-,  then  Ay  ={x  +  Ax)^  -  x^  =  2xAx-\-  Ax^. 
Hence  the  difference  quotient  is 

Ay^2xA^^fA^^2x  +  Aa:. 
A.T  Ax 

We  shall  abbreviate  Difference  Quotient  by  writing  D.  Q. 

Exercises.     Calculate  the  D.Q.  in  the  exercises  of  (205). 

*  /  (x  4-  Ax)  stands  for  the  result  obtained  by  replacing  x  by  x  +  Ax  in  /(x). 


207,208]  DERIVATIVES  181 

207.  The  D.Q.,  -r^,  geometrically.  —  Lot  the  curve  in  tlie  fig- 

y 

ure  represent  a  part  of  the  graph  of  the 
equation  y  =  f{x). 

Let  P  be  a  point  on  the  curve  hav- 
ing coordinates  (x  =  OM,  ij  =  MP), 
and  P'  a  second  point  {x  -{-  Ax  =  OM', 
y-{-Aij  =  M'P'). 

Let  the  secant  PP'  make  an  angle  6'  with  the  rc-axis. 

Draw  PQ  II  OX.     Then  from  A  PQP', 

tan^'  =  ^. 
Ax 

Slope.  —  The  tangent  of  the  angle  which  a  line  makes  with  the 
rc-axis  is  called  the  slope  of  the  line. 

Hence,  the  difference  quotient,  -v-;,  is  the  slope  of  the  secant  drawn 

through  the  points  (or,  y)  and  (x  +  Ax,  y  +  Ay). 

208.  Limit  of  D.  Q.  =  Slope  of  Tangent.  —  Let  the  point  P' 
move  back  along  the  curve  and  approach  the  point  P.  Then  Ax, 
and  in  general  also  Ay,  approach  0. 

i  Suppose  now  that  as  Ax  approaches  0  the  D.  Q.  approaches  a 
definite  limit,  m. 

Then  the  line  through  the  point  {x,  y)  having  the  slope  m  is 
called  the  tangent  to  the  curve  y  =  f{x),  {x,  y)  being  the  point  of 
contact. 

In  the  figure,  as  P'  approaches  P,  the  secant  line  PP'  gradually 
rotates  about  P  and  approaches  a  limiting  position  PT,  which  is 
defined  to  be  the  tangent  to  the  curve  at  P. 

If  d  be  the  angle  which  the  tangent  to  the  curve  at  P  =  {x,  y) 
makes  with  the  x-axis,  then 


^  X      (read,   ''tangent  of   6  equals  the 


tan  e  =^ljm|^j       jj^^  ^^  A|  ^^  ^^  approaches  0." 

When  —  approaches  a  definite  limit  a  tangent  is  thereby  deter- 
Ax     ^^ 

mined.     When  such  limit  is  indeterminate,  the  tangent  does  not 

exist,  or  several  tangents  may  be  drawn  at  P.     We  shall  consider 

only  cases  where  a  single  determinate  tangent  exists. 


182 


DERIVATIVES 


[209 


y  =  X- 
tanO  =  2x 


209.   Examples. 

1.    y  =  X-. 

y  +  Ay={x  +  Aa;)2 

=  x2  +  2  a;  Ax  +  A?. 
Ay  =  2x  Ax  +  A? 

and  ^  =2x+Ax. 
Ax 

Hence        lim    —  =  2  x  =  tan  9. 

Ax^O  ^^ 

Here  the  slope  of  the  tangent  at  any  point 
equals  twice  the  abscissa. 


2.   y  =  ^\xK 

y  +  Ay  =  2V  (x  +  Ax)3 

=  2V  (x'  +  3  x2  Ax  +  3  X  A?  +  Ai^). 
Ay  =  ii  (3  x2  Ax  +  3  X  Ax2  +  Ax^) 

'and      ^  =  ^V(32:2  +  3xAx+ Ax^). 
Ax 


rt 


Hence 


lim  v^  =  na;2  =  tan  0. 


y  =  5V  a;3 
tan  e  =  1x2 


and 


3.   2/  =x2  -2x. 

y  +  A?/  =  (x  +  Ax)2  -  2  (x  +  Ax) 

=  x2+2xAx+Ax^-2x-2Ax 
=  x2-2x+  (2x-2)Ax  +Ax^. 

Ay  =  (2  X  -  2)  Ax  +  A? 
Ay 


=  (2x-2)4-Ax. 


Ay 


y  =  x2  —  2  X 
ton  0  =  2  X  -  2 


lim  — ^  =2x  -2  =  tan». 
Az^oAx 

4.    y2  =  X.     Here  y  is  an  implicit  func- 
tion of  X.     Solving,  we  have 

y  =  ±'s/x. 


The  upper  sign  gives  that  part  of  the  curve  lying  above  the  x-axis,  the 
lower  sign  the  part  below  the  axis.    We  consider  first  the  upper  sign  only. 


209] 


DERIVATIVES 


183 


Then 
y  =  V-c 


and     y  +  Ay  =  Vx  +  Ax. 
A(/  =  \Jx  +  Ax  -  V-C- 


Multiplying  and  dividing  by 


Ay 


V-c  +  Ax+  Ax,  we  get 
( Vx  +  Ax  -  Vx)  ( Vx  +  Ax  +  Vx) 
\Jx  +  Ax  +  '^x 
Ax 


ton  fl  =  ± 


Vx  + Ax  +  Vx 

Hence  ^  = -=L ^, 

Ax       V^  +  Ax  +  V^ 

lim  --^  = ~  =  tan  9. 

Ax-^oAx^  2VaJ 

For  the  lower  part  of  the  curve,  replace  Vx  by  -sjx. 

5.   x2  +  y2  =  100. 

Solving  for  y,  we  get 


2Va; 


and 


ij=±  VlOO  -  x2. 

Considering  first  only  the  upper  half  of 
the  circle  (figure)  we  have 


VlOO  -  (x  +  Ax)2 


y  =  VlOO  -  x2  ; 

y  +  Ay 
.-.    Ay 

Multiplying  and  dividing  by  the  sum  of 
the  two  radicals, 


VlOO  -  (x  +  Ax)2  -  Vioo  -  x2. 


VlOO 


Ay 


■2x  Ax  -  Ax2 


Hence 
and 


VlOO  -  (x  +  Ax)2  +  VlOO  -  x2 

Ay  _      2x  +  Ax 

Ax 


lim 


Ay 


VlOO  -  (x  +  Ax)2  +  VlOO 
2x  X 


=  tan  6. 


X2 


Ax=^oAx         2  VlOO  -  x2  VlOO 

At  any  point  on  the  lower  half  of  the  circle,  tan  d  =-\ .  • 

VlOO  —  x2 

In  all  these  examples  the  slope  of  the  tangent  at  any  given  point  may  be 
obtained  by  substituting  the  abscissa  of  the  point  in  the  value  of  tan  0. 

Exercises.     Calculate  the  slopes  of  the  tangents  at  any  point  (x,  y)  on  the 
following  curves: 

1.  y  =  I  x3.  4.    2/2  =  4  X.  7.    x^  -  y^  =  1. 

2.  y  =  2  x2  -  3  X.  5.   y2  =  -  9  X.  8.   9  x^  +  16  y'  =  144. 

3.  y  =  x3  -  X.  6.    x2  +  y2  =  1.  9.    4  x2  -  y2  =  4. 
Calculate  the  slope  in  each  of  these  examples  when  x  =  1.     Note  the 

results  in  exercises  6  and  7  and  explain. 


184  DERIVATIVES  [210,211 

210.   Derivative.  —  The  expression  lim  (  -^j  occurs  so  frequently 

in  mathematics  that  a  special  name  is  applied  to  it.  Starting 
with  y  as  any  given  function  of  x,  say  /  (x) ,  we  can  derive  from 
this  a  second  function  of  a;  as  follows.  Calculate  f  (x  -{-  Ax)  —  f  (x) 
or  Ay,  divide  by  Ax,  and  pass  to  the  limit  by  allowing  Ax  to  approach 
zero.     Call  the  new  function  of  x  so  obtained  f'{x),  so  that 


/'«=i™(i!)- 


This  is  called  the  ^rs^  derived  function  off{x)  or  thej^rs^  derivative 
off(x),  and  the  expression 


sx^oXAxJ 


is  called  the  first  derivative  of  y  with  respect  to  x.     It  is  usually 
written  in  one  of  the  forms 

Hence  the  slope  of  the  tangent  to  the  curve  y  =  f(x)  at  a  point 
(x,  y)  is 

dv 
doc 

211.  Calculation  of  Derivatives.  —  We  have  already  calcu- 
lated the  derivative  of  y  with  respect  to  a;  in  a  number  of  cases. 
We  now  obtain  a  few  simple  formulas  for  the  calculation  of  deriva- 
tives. Three  steps  are  involved  in  every  case:  (1)  the  calculation 
of  Ay,  (2)  division  by  Ax,  (3)  evaluation  of  the  limit  as  Ax  =  0. 
We  shall  assume  that  such  a  limit  exists. 

Formulas  for  Calculating  Derivatives. 
I.    J)^  (c)  =  0,  c  being  a  constant. 

(1)  For  if  c  is  a  constant  its  change  is  0,  hence  Ac  =  0. 

(2)  Therefore  ~  =  0. 

Ax 

(3)  Hence  lim  ^^  =  0     or     D,  (c)  =  0. 

Ax  =  0  Ax 


211]  DERIVATIVES  185 

II.  D^  {cij)  =  c  D^i/,  c  being  any  constant. 
Proof. 

(1)  The  increment  in  y  being  Ay,  the  increment  in  cy  will  be  c  Ay, 

(2)  Dividing  by  Aa:,  the  D.  Q.  of  cy  relative  to  x  is  c-r^- 

Ax 

Av 

(3)  Let  Aa:  =  0.     Then  c  does  not  change,  while  — ^  becomes 

Dx  (y) .     Hence 

DAcyy=Vimc^  =  cD,y. 
Ar=o  Aa; 

III.  When  ?/  is  a  sum  of  several  functions  of  x,  as 

y  =  u-{-v-\-w-\-  •  •  •  ,    where    u,  v,  w,  .  .  . 
are  functions  of  x,  then 

n^u  =  D^ti  +  n^v  +  D^w  +  •     •   . 

Proof.  When  x  takes  an  increment  Ax,  let  the  corresponding 
changes  in  u,  v,  w,  .  .  .  he  Au,  Av,  Aw,  .  .  .  respectively.  The 
total  change  in  y  is,  therefore, 

(1)  Ay  =  Au  -\-  Av  -]-  Aiu  -\-  ■  ■  •  . 

(2)  Then  ^- =  f^ +  ^ +  p  +  .  .  .  . 

Ax      Aa;       Aa:      Aa; 

(3)  Let  Ax  =  0.  Then  by  definition  (210),  ^  approaches  D^y, 
rr-  approaches  D^u,  etc.     Hence 

D^y  =  D^u  +  DxV  +  D^w  +  •  •  •  ,  when  y  =  u  -{-v  -\- w  -\-  ■  ■  •  . 

IV.  Let  y  be  the  product  of  two  continuous  functions  of  x, 
say  u  and  v. 

y  =  u  '  V. 

When  X  is  changed  to  a;  +  Ax,  let  u  change  to  u  +  Am  and  v  to 
V  -}-  Av.     Then 

2/  +  A?/  =  (u  +  Au)  {v  +  Av)  =  ?^y  +  M  At'  +  y  Af<  +  Am  Ay. 

(1)    Hence  Ay  =  u  Av  -\- v  Au  -\-  Au  Av. 


(2)    Then  -^  =  -u-_  +  y-— 4-Au  — • 

Ax         Ax         Aa;  Ax 


186  DERIVATIVES  [212 

(3)    Let  Ax  =  0.     Then  t^'  -;— '  -r~  approach  D^y,  D^u  and  D^v 

^    ^  LX     LX     diX 

respectively.     Also  Aw  =  0,  since  we  assume  m  to  be  a  continuous 
function  of  x  (205).     Hence  (2)  becomes 

D^y  =  u  D^v  +  V  Da^u,  when  y  =  u  •  v. 

V.   Let  y  =  -,u  and  v  being  continuous  functions  of  x. 

u  -\-  Aw 
Then  y  +  C.y  =  -^^^-^, 

u-\-  Ml      u      V  iya  —  u^v 

(1)  and  Ay  =  -^^  -  -  =     ,2  +  ,Ay  " 

Am  Ay 

V- U-— 

Av         Ax         Ax 

(2)  Hence  ^^  =    „.  +  „a, 

A?/       t^D^ii  —  uD^t^ 

(3)  and  I>.2,=Jim-= ;, 

VL   Let  y  be  a  function  of  u,  where  m  is  a  function  of  x.     Thus 

y  =  w2^2m;  m  =  2a;2  +  l. 
When  x  changes   to  re  +  Ax,  u  changes  to  m  +  Aw  and  y  to 

Ay  _  Ay     Aw 
Now  A^-Aw'Ax' 

Hence  I>^y  -  1>„2/  •  D^u. 

Collecting  our  formulas  we  have: 

(A)  D^c  =  0. 

(B)  D^  {cy)  =  c  I>^?/. 

(C)  i>x  («  +  t'  +  ^t'  +  •    •    •  )  =  I>ocU  +  I>xi^  +  DooW  +  •    •    •   . 

(D)  n^  (u  '  v)  =  u  D^v  +  ^  I>a>u. 

/ii\       vD^ii  —  uD^v 

(F)  D^y  =-  D.,y  '  n^u. 

212.   We  next  derive  the  following  standard  formulas: 

(G)  i/=x";  DJ/  ^nx'^-K 
(H)  i/-logx;  !>.?/  =  ^- 


212]  DERIVATIVES  187 

(I)  y  =  r/';  n^!/  -  a*  log  a. 
(J)  2/  =  sinx;  D^y  =  cos  a?. 
(K)   y  =  cosir;  D^y  =  —  sinx. 

(G)  y  =^  x";         assume  n  to  be  a  positive  integer. 

(1)  Hence  A2/  =  nx"-iAx  +  ''^^^^^^a:''-2Ai'+  •  •  •  +  aI-". 

(2)  Then    ^  =  nx" -  +  "^^f^ ." -' A.  +  •  •  •  +Ax"-'. 
^  ^  Ax  1*2 

(3)  Let  A.T  =  0.     All  terms  on  the  right  of  the  last  equation 
vanish  except  the  first,  and 

lim  ^  =  D,y  =  nx^-K 

The  proof  when  n  is  not  a  positive  integer  will  be  given  after 
formula  (H)  is  derived. 

(H)  y  =  logx;         ?/  +  A?/  =  log  {x  +  A.t). 

y-  _J_  At  /  At  \ 

(1)    A2/ =  log  (x  +  Ax)  -  log  X  =  log — ^-  =  log(^l+-^j- 

(3)    Let  Ax  =  0.     We  must  evaluate 

X 

'  I         Ax 
lim  log  I  1  +  -^ 

Ax  =  0  \  X 

Let  z  =  — ;  then  2  =  20  when  Ax  =  0,  provided  x  ?^  0.     [x  =  0 
Ax 

is  excluded  by  our  standing  assumption  of  continuity  (205).]     We 

must  now  evaluate 

Let  z  =  1,  2,  3,  .  .  .  ,  n.     The  corresponding  values  of  fl  +  -j 
are  2,2.25,2.37,  ...,(!  +  -).     As  n  increases,  these   values 


188  DERIVATIVES  [212 

steadily  increase,  but  always  remain  less  than  3,  no  -matter  how 
large  n  may  be.     For,  by  the  Binomial  Theorem, 

y-^n)   ~^'^'\^     1-2        n2+        1.2-3  n^^ 

to  (n  +  1)  terms 

_■,.,,  }~n)      V~7ilV~^il  ,  ^to(n  +  l) 

■^"^1-2^        1.2-3        "^  \  terms. 

As  n  increases,  each  term  of  the  expansion  increases  as  well  as 
the  number  of  terms.  Also  all  the  terms  are  positive.  Hence 
their  sum  increases  with  n.  Further  compare  the  above  expansion, 
leaving  out  the  first  term  (=1),  with  the  geometric  progression 

2       2-  2"  ~  ■*■ 


whose  sum  is  less  than  2. 


(-^■) 


For  all  values  of  n,  however  large,  our  expansion  is  less,  term  for 
term,  than  the  progression.  As  n  =  oo ,  the  sum  of  the  progression 
approaches  2,  hence  the  expansion,  excepting  its  first  term,  ap- 
proaches a  limit  less  than  2.  Adding  the  first  term,  the  limit  is 
less  than  3. 

This  limit  is  an  irrational  number  denoted  by  the  letter  e,  and 
has  the  approximate  value 

e  =  2.7182818  +  •  •  -  . 

We  have  now  the  result  that 

1 


lim   1  + 


z 

when  z  approaches  infinity  through  positive  integral  values.  The 
same  is  true  when  z  increases  continuously,  but  we  shall  not  stop 
for  the  proof,  which  may  be  found  in  texts  on  the  calculus. 

Then  lim  log  ( 1  +  - )  =  log  e, 

and  hence  Z>x  (log  x)  =  ^  log  e. 


212]  DERIVATIVES  189 

Let  us  now  take  e  as  the  base  of  our  system  of  logarithms,  so  that 
log  X  shall  mean  loge  x.     Then 

loge  =  logeC  =  1. 
Hence  D^  (log x)=  -- 

Logarithms  to  the  base  e  are  called  natural  or  Naperian  loga- 
rithms. In  the  theory  of  mathematics  natural  logarithms  are  in 
general  use,  common  logarithms,  to  the  base  10,  being  utilized  only 
for  numerical  computation. 

We  can  now  derive  formula  (G)  without  any  restriction  on  the 
value  of  n. 

From  y  =  x"" 

we  have  logy  =  n  log x.     (Base  e.) 

Hence  Z)x  (log  y)  =  Dj.  (n  log  x). 

Now  in  formula  (F)  replace  y  by  log  y  and  u  by  y.     It  becomes 

D,(log  y)  =  D,(log  y)  •  D,y  =  \^D,{y),  from  (H). 
Also  Dx{n\ogx)  =  ->    from  (B)  and  (H). 

-D^y  =- 
y  X 

or  Dxy  =  —-  >    where  y  =  x". 

X 

Hence  D^x""  =  —  =  nx""-^. 

X 

(I)  y  =  a^. 

Taking  logarithms,    logy  =  x  log  a. 
Hence  D^  (log  y)  =  D^  (x  log  a). 

But  Dx  (log  y)  =  -Dxy  (see  above) 


and  Dx  (x  log  a)  =  log  a. 

Hence  -  D^y  =  log  a, 

or  D^y  =  y  log  a,  where  y  =  a*. 


190  DERIVATIVES  [213 

Therefore  Z)jO'^  =  a""  log  a. 

(J)  y  =  B\nx',        y-\-  Ay  =  sin  {x  +  Ax). 

I         Ax\        Ax 
(1)      Ay  =  sin  (x  +  Ax)  —  sin  a:  =  2  cos  ix-{-  —  j  sin  -^  •  (158.) 

/     ,    Ax\   .Ax  .Ax 

2  cos  kc  +  -^    sm  -p^  -         a    x  sm 


Ay      -^--V-^T;-T  /        A.N-2 

(2^      Ax-  ^^  ^°'V    +2;     A^ 

2    . 
(3)     Let  Ax  =  0.     Then 

.    Ax 
sm  — 

cos  fa:  +  ~]  =  cos  x,  and  — ^ —  =  1.     U60.  Replace  x  by^-j 


lim  --^  =  1>^  sin  a?  =  cos  a?. 

AX  =  0  Arr 

(K)  y  =  cosx;        ?/  +  A^/  =  cos  (x  +  Ax). 

/         Ax\        Ax 

(1)    A?/  =  cos  (x  +  Ax)  -  cos  x  =  -  2  sin  f  x  +  -^  j  sin  y  •     (158.) 

.    Ax 

.  sm  -^ 

Ay  .    /     ,    Ax\         2 

(2)  Ax  =  -^^n"+T)-^- 

2 

(3)  .'.         lim  -^  =  J)^  cos  a?  =  -  sin  a;, 

^  ^  Ax^o  Ax 

By  suitable  combinations  of  formulas  (A)  to  (K)  the  derivative 
of  any  function  may  be  calculated. 

213.     Examples. 

1.   Calculate  Dxi^x^  +  ^x). 

Dx  (4  x3  +  3  x)  =  Dx  (4  x3)  +  Dx  (3  x)     (C) 
=  4  Dxx^  +  3  Dxx  (B) 

=  12  x2  +  3.  (G) 


2.    Calculate  ^^(l+bgx) 


214,215]  DERIVATIVES  191 

J.   (       c^        \  _  (1  +  log  x)  D:ce'  -  fi^Dx  (1  +  log  x)      .„. 
"^'[l+logxl  (1+logx)^  ^""^ 


(D,  (C),  (H), 


(1  +  log  x)  e^  —  e^  - 

'       (r+iogx)2 

^      xd+logx)-! 

X(l+l0gx)2      • 

3.   Calculate  DxCSsin^x). 

Dx(3sin2x)  =3Dxsin2x     (B) 

=  6  sin  X  Dj;  sin  x     (F) ;     {u  =  sin  x) 
=  6  sin  X  cos  x. 

214.   Exercises.     Calculate  D^^y  when : 

1.  y  =  3  x4  +  5  x3.  10.    ?y  =  log  (x  +  2). 

2.  y  =  2-3  +  1.  11-    2/ =  log(3x2  -  1). 

Q  1    J    ,    1     i  12.    7/  =  f^  log  X. 

»  3.    t/  =  ^x'  +ix*. 


4.    y  =  x    =  -  2  x^. 


13.    2/ =  sin  X  log  cos  X. 


,  1  14.    2/  =  esinx. 

5.  2/  =  -L+_L. 

Vx      v^; 

6.  y  =  sinx  +  e^^. 

7.  w  =  e^.  16.    y  =  cotx 


I  c            i         /      sin  X 
15.    y  =  tan  x    = 

V      cos  X 


,8.    y  =  a^^  17.    y  =  log  tan  X. 

18.   y  =  sec  x, 

215.   The  Derivative  as  a  Rate  of  Change.  —  The  difference 

Aw 
quotient  — ^  gives  the  average  rate  of  change  of  y  relative  to  x  when 

X  changes  by  an  amount  Ax.  The  smaller  Ax,  the  more  nearly  will 
the  D.  Q.  represent  the  actual  (or  instantaneous)  rate  of  change 
of  y  relative  to  x.  Hence  the  limit  of  the  D.  Q.  as  Ax  =  0  is  taken 
as  the  actual  rate  of  change. 

Rule.    To  find  the  rate  of  change  of  one  quantity  relative  to  another, 
calculate  the  derivative  of  the  first  quantity  with  respect  to  the  second. 

Examples. 

1.  y  =  x2.     Then     Dxij  =  2  x. 

Hence  y  changes  2  x  times  as  fast  as  x. 


192  DERIVATIVES  [216,  217 

2.  In  the  case  of  a  falling  body,  if  s  be  the  space  and  t  the  time  and  the 
body  starts  from  rest,  we  have 

s  =  I  gt^. 

Then  Dis  =  gt  =  velocity  at  time  i. 

3.  Find  the  rate  of  change  of  the  volume  of  a  sphere  relative  to  the  radius. 

F  =  |7rr3;  DrV  =  4irr2. 
That  is,  the  volume  of  a  sphere  changes  4  irr2  times  as  fast  as  the  radius. 

216.  Exercises.     Calculate  the  rate  of  change  of: 

1.  y  relative  to  x,  when  y  =  x^  +  x^. 

2.  y  relative  to  x,  when  y  =  sin  x. 

3.  y  relative  to  x,  when  y  =  sin  x  cos  x. 

4.  y  relative  to  x,  when  y  =  sin2  x  +  cos^  x. 

5.  y  relative  to  x,  when  y  =  e^. 

6.  the  volume  of  a  cube  relative  to  its  edge. 

7.  the  surface  of  a  cube  relative  to  its  edge. 

8.  the  surface  of  a  sphere  relative  to  its  radius. 

9.  the  volume  of  a  cylinder  relative  to  its  altitude. 

10.  the  volume  of  a  cone  relative  to  the  radius  of  its  base.  , 

11.  the  area  of  a  circle  relative  to  its  perimeter. 

12.  A  body  starts  when  t  =  0  and  moves  so  that  the  space  described  in 
time  t  (seconds)  is  s  =  16  <2+  10.     Find  its  velocity  when  t  =  10;  t  =  5;  t  =  0. 

13.  The  space-time  equation  being  s  =  2t^  +  3t  —  5,  find  the  velocity  at 
any  time  /;  what  is  it  when  t  =  10;  i  =  1;  i  =  0  ? 

14.  As  in  13,  when  s  =  10  sin  ( 3  t  +  |j. 

16.  Given  two  sides  and  the  included  angle  of  a  triangle.  Calculate  the 
rate  of  change  of  the  third  side  relative  to  each  of  the  given  sides  and  to  the 
given  angle. 

217.  Higher  Derivatives.  —  When  i/  is  a  function  of  x,  D^y  is  in 
general  a  new  function  of  x;  the  derivative  of  this  new  function 
is  called  the  second  derivative  of  y  with  respect  to  x  and  is  written 
Dly.  The  derivative  of  the  second  derivative  is  called  the  third 
derivative,  written  D^y,  and  so  on. 

'    Exam-pies. 

1.  2/  =  x3.         Dxy  =  3x2;        D|2/  =  6x;  D^y  =  & ;        Dly  =  0. 

2.  y  =  smx.     Dxy  =  cos x;       Dly  =  — sin  x;     D^?/ =  —  cosx  ;  etc. 

3.  y  =  X".         Dj2/  =  nx«-i;     D^y  =  n  (n  —  l)x"-2  ;  .... 

D>  =  ?i  (n  -  1)  .  .  .  1  =  |n. 


218]  MACLAURIN'S   SERIES  193 

218.    Maclaurin's  Series.  —  Suppose  that  a  given  function  of 
X,  f  {x) ,  can  be  represented  by  a  converging  power  series  in  x,  thus : 

(1)  f  {x)  =  Co -\- cix  +  cox"^ -{- czx^ -{■  '  ■  ■   +c„a;"+   •  •  •    . 

To  find  the  values  of  the  coefficients  Cq,  ci,  C2  ■  ■  •    .     Put  x  =  0 
in  (1)  and  we  have  Cq  determined  by 

/(0)=co. 

To  get  Ci,  calculate  DJ{x)  or  f'(x)  from  (1); 

(2)  f'(x)  =  ci  +2c2X  +  3c3X-  +  •  •  •   +  /ic„x"-i  +  .  .  .    . 

Put  X  =  0  in  (2)  and  we  have  ci  determined  by /'(O)  =  cu 
From  (2)  calculate  DJ'(x)  or  f"{x); 

(3)  /"(x)=2c2  +  2.3c3a;4-  •  •  •   +w(n- l)x"-2+  .  .  .    . 
Put  a:  =  0  in  (3)  and  we  have 

r(0)=2c2     or     C2  =  ^r(0). 
Calculating  DJ"{x),  or  f"'{x),  we  have 

(4)  /'"(a:)  =  2  . 3  C3  +  •  •  •  +  n  (n  -  l)(n  -  2)  a;"-^  +  •  .  .    . 

1 


When  X  = 

0, 

/' 

"(O)- 

=  2.3c3 

;    C3 

2 

~/'"(0) 

Similarly, 

C4  = 

=  2-. 

1 

3.4 

r(o)= 

(0), 

^  /^"H0)=  ,^-/^"K0). 


n  (n  —  1)  .  .  .  1  •'  1 7i 

Hence 

/(x)  =  /(0)+a:/'(0)  +  ^r(0)  +  ^'r'(0)+  •  •  •  +fVn)(o)+  .  .  .  . 

Here  f"\0)  is  found  by  differentiating /(x)  n  times  in  succession 
and  putting  x  =  0  in  the  result. 

The  above  result  is  called  Maclaurin's  series  for  the  function 
fix).  In  obtaining  it  we  have  tacitly  assumed  that,  if  f{x)  be 
represented  by  a  power  series,  the  derivative  f'{x)  can  be  calcu- 
lated by  differentiating  the  series  term  by  term. 


194  MACLAURIN'S  SERIES  [219 

219.    Examples. 

1.   Develop  e^  in  a  power  series  in  x. 

fix)  =  e^;    nx)  =  e^',    /"(x)  =  e^;  .  .  .  ;  }^^\x)  =  e^. 
Putting  X  =  0,  we  have 

/(O)  =  1 ;    /'(O)  =  1 ;    /"(O)  =  1 ;  .  .  .  ;  /<"H0)  =  1. 


Hence 


->+-+|+i+---+|+ 


This  series  converges  for  all  values  of  x,  and  is  used  for  calculating  the  value 
of  e^  to  any  desired  degree  of  approximation. 
When  X  =  1, 

from  which  e  can  be  found  approximately  by  taking  a  few  terms  of  the  series. 

2.    Develop  sin  x  in  a  power  series  in  x. 

/(x)  =  sinx;    /'(x)=cosx;    /"(x)  =  -sinx;    /'"(x)  =  -  cos  x,  .  .  .    . 
When  X  =  0, 

/(0)  =  0;    /'(0)=1;    /"(0)=0;    /'"(O)  =  -  1,  etc. 
Hence 

X^        X**        x^ 

sinx=x-|3+^-|y+-  •  •    . 

This  series  converges  for  every  value  of  x,  and  may  be  used  for.  finding  sin  x 
to  any  degree  of  approximation.     Thus,  put 

X  =  10°  =  :^  radians. 
Then 


^'''^^°^h>-\[v^'-^m[l^'- 


Note.  In  computing  mth  an  alternating  series  {signs  alternately  +  and  — ), 
the  error  committed  in  rising  only  a  few  of  the  first  terms  of  the  series  is  always 
numerically  less  than  the  first  term  neglected. 

Thus  the  error  in  sin  10°  as  obtained  from  the  three  terms  written  above  is 

less  than 

A^  fiLV  or  less  than  .000  000  000  98. 
5040  \  18/ 

Hence  the  error  is  less  than  1  unit  in  the  ninth  decimal  place. 
Exercise.     Show  that 

cosx  =  l-|-2+||-^+  •  •  •   . 
Calculate  cos  10°  to  five  places. 


220]  BINOMIAL  THEOREM  195 


3.    Develop  log  (1  +  x)  in  powers  of  x. 

/•(j)=log(l+x); 

/(O)  =  logl  = 

^'^^)=i+x: 

/'(0)=1. 

f"(T\  —                          •'■ 

/"(0)  =  -l. 

^^^)-         (1+X)2' 

t"'(^\  —           " 

/"'(0)=2. 

^     ^^)-(l+x)3' 

—  2-3 

^•^(^)=(l+x)-- 

/'^•(0)  =  -2.3. 

log(l  +  x)=x-|+|' 

-?+ ■'■•.■ 

This  series  converges  only  when  —  1  <  x  =  1 ,  and  hence  can  be  used  only 
when  X  lies  between  —  1  and  + 1  and  for  x  =  + 1 . 

Since  the  base  of  the  logarithm  system  in  log  (1  +  x)  is  under- 
stood to  be  e,  the  last  series  enables  us  to  calculate  the  natural 
or  Naperian  logarithms  of  numbers  from  0  to  2,  exclusive  of  0. 
For  1  +  X  ranges  from  0  to  2  when  x  ranges  from  —  1  to  + 1 .  In 
particular,  when  x  =  1  we  have 

log,  2  =  l-i  +  i-i+-... 

This  is  a  convergent  alternating  series.  Since  in  such  a  series 
the  error  committed  by  neglecting  all  terms  after  a  given  one  is 
less  than  that  term  (199)  *,  1000  terms  of  the  series  would  be  required 
to  give  log  2  correct  to  three  decimal  places.  The  series  therefore 
converges  too  slowly  for  practical  use.  A  more  serviceable  series 
will  be  considered  in  the  next  chapter. 

220.  The  Binomial  Theorem.  —  When  n  is  a  positive  integer,  we 
have 

(l+;r)"  =  l  +  na:  +  '^^^^^a:2+  •  •  •  +  x". 

We  shall  now  derive  the  formula  for  expanding  (1  +  x)"  in 
powers  of  x  for  any  value  of  n,  positive  or  negative,  integral  or 
non-integral. 

Let  /(x)=  (l+:r)". 

*  Apply  (199)  to  the  neglected  part  of  the  given  series. 


196  BINOMIAL  THEOREM  [220 

Then 

/'(x)=n(l+rc)'»-i;  /'(0)=n. 

fix)  =  n  (n  -  1)  (1  +  a:)"-2;  /"(O)  =  n  (n  -  1) 

J"'{x)  =n{n-  1)  (n  -  2)  (1  +  a;)"-^;  /'"(O)  =  n  (n  -  1)  (n  -  2). 

/""'  {x)=n{n-l){n-2)  .  .  .  (n  -  m  +  1)  a;"-"*; 
/'"»>  (0)  =  n  (n  -  1)  (n  -  2)  .  .  .  (n  -  w  +  1). 

Hence  by  Maclaurin's  series, 

,  n(n  —  l)     ^   ,   n(n  —  1)  (n  —  2)     „    , 
(1  +  ic)"  =  1  +  na?  +  -J-^^'  +  1.2.3 "^    +  *   "   ' 

n  (rt  -  1)   .   ♦   '   (n  -  m  +  1)     «.,... 
"^  1  .  2  ....  m  -T       •   •  > 

provided  that  the  serfes  on  the  right,  called  the  Binomial  Series, 
converges. 

Convergence  of  the  Binomial  Series.  —  Denote  the  mth  term 
of  the  series  by  u^,  the  (w  +  l)th  term  by  w^+i-     Then 
_  n  (n  -  1)  (n  -  2)  .  .  .  (n  -  m  +  2)    ^_^ 
^"^  ~  1  .  2  .  3  ....  (w  -  1)  ^        ' 

n  (n  -  1)  (n  -  2)  .  .  .  (n  -  m  +  2)  (n  -  m  +  1) 
1.2.3-  .  .  .  (m  -  1)  .  m 


Um+l   =  — '   \       ^\ ^-Tin TV^. ^" 


Applying  the  ratio-test  (202),  we  have 

u^  ^  n-m  +  1  ^  ^  /n±l  _  ^  ^_ 
u^  m  \    m  J 

The  quantity  in  the  last  parenthesis  is  numericallij  less  than  1, 
when  m  is  larger  than  w  +  1 ;  to  secure  this  we  simply  start  far 
enough  out  in  the  series  to  make  m  >  n  +  1.  Then  the  ratio 
Um  +  \  -^  Um  will  be  numerically  less  than  x,  and  hence,  if  x  he 
numerically  less  than  1,  the  series  converges.  When  x  is  numeri- 
cally greater  than  1,  the  series  diverges.     For  the  ratio  m^+i  -^  Um 

equals  the  product  of  two   factors,  ( —  Ij  and  x.     As  m 

increases  the  first  factor  approaches  —  1  as  a  limit.  Hence  if 
|a:|  >1,  the  product  will  also  ultimately  be  greater  than  1  numer- 
ically. Finally,  when  a;  =  ±  1  our  binomial  reduces  to  2"  or  0 
respectively  and  we  need  not  consider  the  series  at  all. 


2211  BINOMIAL  THEOREM  197 

We  therefore  use  the  binomial  series  for  (I  +  x)"  only  when 

|x|<l. 

221.   Binomial  Series  for  («  +  6)".  —  We  have 
bY 


(a  +  br  =  a"  (l  +  ^J 

A,     b     n(n-l)b-  n(n-l)-.-(n-m+l)6'"  ,        \ 

1,  "^''a^    1-2      a-'"^""^  1.2-...m  a'^^'") 


or, 

(a  +  6)"  =  a"  +  n«»-i  6  +  !il^_zil,,«-2ft5  ^  .  .  . 

,   n  (n  -  1)  .  .  .  (»  -  m  +  l)     „_„.    ,„ 
"^  1  . 3  ....  m  -r  •  •  •  . 

The  series  converges  when    -    <  1,  that  is,  when  b  is  numerically 

less  than  a. 

The  mth  term  of  the  expansion  is 

y    _n{n-l)  .  .  .  (n-m  +  2)  _^ 

"""•  ~         1  .  2  ....  (m  -  1)         ""  ■  • 

Examples. 


2      '       1-2  1-2.3 

=    1   -^,X-IX^-:(\X3+       •    •    •    . 

2.  Find  an  approximate  value  of  V-QS. 

V^  =  Vl  -  .02  =  1  -  ^  (.02)-  i  (.02)2  _  .  .  .  =  .990+. 
The  neglected  part  of  the  series  is  less,  term  for  term,  than  the  G.  P., 
(.02)2  +  (.02)3  +   .   .   .   +  (.02)'^  +   •  •  •  , 
whose  sum  is 

S  =  Y&^  =  .0004  approx. 

3.  Find  the  7th  term  of  the  expansion  of   v  (2  —  3  ^Jx)*  in  powers  of  x. 

%/(2  -  3  V^)^  =  (2  -  3  \/x)K 
Hence  a  =  2,  6  =  —  3  \/x,  n  =  |,  ?n  =  7. 

Then  .,-^'^-\'.<|:3^.';;V'-°'2'-'(-3V.)°-^x.. 

In  this  case  the  expansion  converges  if 

|3  V^l  <2,    or  |9x|  <4,    or    |z|<  t 
For  negative  values  of  x  the  expansion  would  involve  imaginary  terms  be- 
cause of  the  presence  of  yx. 


198  I^XERCISES  [222 

222.  Exercises.  —  Write  the  first  four  terms  of  the  develop- 
ments in  series  of  the  following  functions,  and  give  the  values  of 
X  for  which  the  series  converge. 

1 


1. 

tan  X. 

2. 

secx. 

3. 

sin*  X. 

4. 

sin  x2. 

6. 

e2^. 

6. 

e-'. 

7. 

e^+e-*. 

X 

8. 

eo. 

9. 

X 

e    «. 

10. 

I                    X 

e^  +e    a. 

11. 

sin  X  +  cos  X. 

12. 

sin  ax. 

13. 

vr+^: 

14. 

Vl  -x. 

xu. 

Vl+a; 

16. 

1 
1+x 

17. 

1+x 

1  -X 

18. 

(l-2x)-|- 

19. 

v/(-^^)' 

20. 

(X2-1)-1. 

21. 

(2-x3)l. 

22. 

(V2-v^rl 

23. 

G-i?- 

24. 

I V3       V2/ 

25. 

(2ai  +  3xi)-i 

26. 

(a^  +3x'')t. 

By  use  of  the  binomial  theorem  calculate  to  three  decimal  places  inclusive 
the  values  of: 

27.  \/lO.  31.  V0.096. 

28.  -s/SO.  32.  -n/802. 

29.  ^/68^  33.  -^624:5. 

30.  -s/1121. 

Calculate  to  five  decimal  places  inclusive  the  values  of: 

41.  e-\ 


34. 

sin  25°. 

35. 

sin  5°. 

36. 

sin  1°. 

37. 

sin  10'. 

38. 

cos  50°. 

39. 

cos  100°. 

40. 

1. 
6* 

42. 

1 

43. 

log  1.1. 

44. 

log  1.2. 

46. 

log  (.75), 

CHAPTER  XIV 

Computation.     Approximations.    Differences  and 
Interpolation 

223.  Remarks  on  Computation.  —  (1)  In  a  series  of  similar 
computations,  perform  similar  operations  together.  If  the  same 
number  is  to  be  added  to  each  of  several  others  write  it  on  the 
edge  of  a  slip  of  paper  and  hold  it  over  or  under  each  number  in 
turn. 

(2)  When  a  result  is  wanted  to  say  three  decimals,  computa- 
tions should  be  carried  to  four  places  so  as  to  avoid  accumula- 
tion of  errors  which  would  vitiate  the  third  place. 

(3)  As  a  general  rule,  4-,  5-,  6-,  and  7-place  logarithm  tables 
will  yield  respectively  not  more  than  4,  5  6,  or  7  significant  figures 
of  a  number. 

(4)  Results  should  be  stated  with  an  accuracy  commensurate 
with  that  of  the  data.  Thus,  if  a  line  be  measured  10  times  to 
0.01  ft.,  the  mean  of  the  10  measures  should  be  given  to  0.001  ft. 
More  than  three  places  in  the  mean  would  be  a  useless  refine- 
ment. Do  not  state  an  angle  to  seconds  when  it  results  from 
computations  which  render  even  the  minute  uncertain. 

224.  Useful  Approximations.  —  Let  the  student  verify  that, 
when  X,  y,  u,  v  are  small  decimals,  we  have  approximately: 

6.1^  =  1+.-. 


1. 

(l+x)(l+y)  = 

\+x  +  y. 

2. 

(1  +  x)  (1  -  2/)  = 

l+x-y. 

3. 

(1  -  X)  (1  -  2/)  = 

l-x-y. 

4. 

ri.--^- 

7. 

(1  +  x)  (1  +  y)   . 

■  • 1    1    . 

(1  +  u)  a  +  v)  . 

— ■  —  1  -h  : 

8. 

(1  +  x)'»  =  1  +  nx. 

Asi 

special  cases  of  (S) 

we  have 

9. 

Vl"+a-=  l  +  hx. 

10. 

Vl-x  =  l  -ix 

. 

199 


200 


APPROXIMATIONS 


[224 


12. 


1  1   ^1 


13.  (1  +  x)2  =  1  +  2  X. 

14.  (1  -x)2  =  1  -2x. 

15.  e-^  =  1+  X. 
More  accurately : 
21.    sin  X  =  X  —'J  x3. 

Exam-pies. 

1.    .987  X  .993  =  (1 


16.  loge  (1  +  X)   =  X. 

17.  logio  (1  +  x)  =  .43  X. 

18.  sinx  =  X  (radians). 
19.'  tan  X  =  X. 
20.  cosx  =  1. 

22.   tan  X  =  X  +  J  x3. 
23.   cos  X  =  1  -  i  x2. 

013)  (1  -  .007)  =  1  -  .013  -  .007  =  .980. 
The  error  is  .013  X  .007  =  .000091. 
1      ^  1 

987 


2. 


1  +  .013  =  1.013. 


.013)  i    =  1  -   ^  (-013) 


1  -  .013 

3.  V^987   =   (1   - 
=  .9935,  correct  to  four  places. 

4.  Find  the  range  of  vision  from  a  point  h  ft, 
above  the  surface  of  the  earth. 

Let  A  be  the  station  of  observation  (figure), 

AB  =  h  ft.,     BC  =  DC  =  R  =  3960  miles. 

Then 

R  =  3960  X  5280  ft. 


yJ{R  +  h)2  -  m  =  V2 Rh+h^=  V2 Rh\    1  + 


n/' 


For  moderate  elevations 
mately. 
Hence 


'2R' 


The  error  in  this  value  of  x  is  -r^  x  approximately. 
4  K 

Exercises.     Calculate  the  approximate  values  of, 
^.  .85X1.12 

.982'  1.15  X  .92 

*•    1.125' 


^975 


2R 
small  and  the  second  radical  =  1  approxi- 

l2Rh  approximately. 
h 


5. 


3.   Vl.20; 


6.    (1.15)2. 


7.  Prove  the  last  statement  of  example  4. 

8.  How  far  can  an  observer  see  from  a  mountain  one  mile  high  ? 

9.  What  is  the  distance  to  the  horizon  as  seen  by  an  observer  on  the  sea- 
shore with  his  eye  6  ft.  above  the  water  level  ?     (Three-mile  limit.) 

10.    If  the  range  of  a  gun  on  a  warship  is  10  miles,  how  high  should  the 
lookout  be  stationed  to  detect  objects  coming  within  range? 


225]  COMPUTATION   OF   LOGARITHMS  20l 

11.  What  is  the  error  in  each  of  the  approximations 

(1)  .  .  .  (23)whenx,  r/,M,  v  =  0.1?    When  x,  y,u,i' =  0.01? 

12.  Calculate  to  four  decimal  places  sin  130°  and  cos  (—  100°).     (Reduce 
to  functions  of  angles  <  45°.) 

13.  Calculate  a  4-pIace  table  of  natural  sines,  from  0°  to  45°,  at  intervals 
of  5°. 

14.  As  in  exercise  13  for  a  table  of  natural  cosines. 

225.   Computation  of  Natural  Logarithms. 

We  have    \og(l +x)  =  x- ~ +  ~ -~ -^  •  •  •  . 

Replace  a:  by  —  a; : 

x^       x^       X* 


log  (1  -  x)  =  -  X       2        3        4 

Hence,        log  (1 -^  x)  -  log  (1  -  x)  =  2^x -^~ +  j -\- 

provided  —  1  <  .r  <  1. 

But  Iog(l+a;) -log(l -x)  =  log^  "^^ 


1-x 


^   ^               1+x      w  +  1  1 

Let  :; = ;     or,    x 


I  -  X         n     '        '  2n+l 

Then  log  (1  +  x)  -  log  (1  -  x)  =  log  (n  +  1)  -  log  n 

and 

Iog(«+I)=log»+2[2^  +  3^2^  +  5l2i+I?  +  - ■  •]• 
By  means  of  this  equation  log  (n  +  1)  can  be  calculated  when 
log  n  is  known.     The  series  on  the  right  converges  rapidly  and 
for  all  positive  values  of  n.     Putting  successively  n  =  1,  2,  3,  ...  , 
we  obtain  in  turn  log  2,  log  3,  log  4,  .  .  .    . 

We  will  now  obtain  an  estimate  of  the  maximum  error  made  in 
stopping  at  any  term  of  the  series. 

Let  A;  =  2  n  +  L 

Then  the  mth  term  of  the  series  is 

1 
^'"       (2  m-  l)A;2'n-i' 

and  the  remainder  of  the  series  will  be 

p    _  1  ,  1  I  1 


(2  m+1)  fc2 '"+1  ^  (2  m+3)  k^  '"+3  ^  (2 m+5)  k^^+^ 


202  COMPUTATION  OF   LOGARITHMS  [225,226 

Then  R^  is  certainly  less,  term  for  term,  than  the  series 

1  fl    I    S    ^    I    -   ■     1-1  1 

since  the  series  between  the  brackets  is  an  infinite  G.  P.  with  ratio 
T2-     Also,  since 

k  =  2n  -\-  1  and  n=l,     .'.   fc  >  2  for  all  values  of  n.     Hence 
1 


and  therefore 
R„,  < 


<2 


(2m  + l)A;^'"+i       (2?^  +  1)  (2n  +  l)2™+i 

If  we  now  include  the  factor  2  which  stands  before  the  bracket 
in  the  equation  giving  log  {n  +  1),  the  total  error  is  less  than 
4 

(2  7W+  1)  (2  71+  l)-^'«+i 

when  log  (n  +  1)  is  calculated  by  using  only  the  first  m  terms  of  the 
series. 

Thus  in  calculating  log  5,  we  have  n  =  5  and  the  error  in  stop- 
ping with  the  mth  term  is  less  than 

4 

(2m  +  l)ll2'"+i* 

4 
Hence  when  m  =  1,  the  error  is  less  than  ^      3;  that  is,  if  we  use 

only  the  first  term  of  the  series,  log  5  will  come  out  correct  to  3 
decimal  places  inclusive.     When  m  =  2,  the  error  is  less  than 

4 
TT-TT^ ,  so  that  the  first  two  terms  will  give  log  5  correct  to  5  places, 

O'lP 

and  so  on. 

Exercises. 

1.  What  is  the  error  in  log  7  when  only  one  term  of  the  series  is  used?    When 
two  terms  are  used  ? 

2.  How  many  terms  of  the  series  are  required  to  give  log  7  correct  to 
10  places? 

3.  How  many  terms  of  the  series  are  required  to  give  log  17  to  20  places  ? 

4.  Calculate  a  four-phice  table  of  natural  logarithms  of  the  numbers  from 
1  to  20  inclusive. 

226.    Common  Logarithms.  —  When  the  natural  logarithm  of 
a  number  is  known,  its  common  logarithm  may  be  found  by 


227]  DIFFERENCES  203 

multiplying  by  a  certain  constant  factor  called  the  modulus  of  the 
common  system  of  logarithms.  We  shall  show  that  this  modulus, 
or  multiplier,  is 

M  =  logio  6  =  0.4342945  .... 

Let  the  natural  logarithm  of  any  number  be  x,  its  common  loga- 
rithm y.    To  express  y  in  terms  of  x.    We  have,  if  n  be  the  number, 

loge  n  =  X     and     logio  n  =  y, 
or,  n  =  e^    and     n  =  10^. 

Hence  10^  =  e\ 

To  solve  for  y,  take  logarithms  of  both  members  to  the  base  10. 
Then  y  =  a:logioe,   . 

which  proves  our  statement.  To  find  the  value  of  logio  e,  we  need 
only  calculate  loge  10  and  take  the  reciprocal  of  the  result. 

Exercises. 

1.  Calculate  the  modulus  Af  to  5  places. 

2.  Calculate  logio  101  to  10  places. 

3.  Calculate  logio  11  to  10  places. 

4.  Calculate  a  four-place  table  of  common  logarithms  of  the  numbers 
from  1  to  20  inclusive. 

227.  Differences. — Consider  a  sequence  of  quantities  uq,  ui, 
U2,  .  .  .  ,  Un,  .  .  .  ,  and  form  the  differences,  Auq  =  ui  —  Uq, 
Aui  =  U2  —  Ui,  .  .  .  ,  Aun-i  =  Un  —  Un-i,  ■  •  ■  ,  callcd  the  first 
differences.  Form  next  the  differences  of  these  differences,  called 
the  second  differences  of  the  original  sequence,  and  so  on.  We 
obtain  in  this  way  the  entries  in  the  following  difference  table, 
where  the  successive  difference  columns  are  denoted  by  Ai,  A2,  A3, 
.  .  .  and  the  original  sequence  by  Aq. 


uo 


Ao 

Ai 

A2 

As 

Mo 

Ml  -  Mo 

U\ 

Uo  —   Ml 

M2  -  2  Ml  +  Uo        ^^ 

-  3  U2  +  3  ?/.i 

M2 

M3  -  M2 

M3  —  2  U2   +  Ml 

U3 

Un-2 

M„_i  -  Un-2 

M„_i 

Un  —  Un-l 

Un   —  2li„_i  +  Un-2 

Un 

204  DIFFERENCES  .  [228 

We  observe  that  the  coefficients  follow  the  binomial  law.  Let 
the  student  prove  by  induction  that  this  law  is  followed  in  all 
the  successive  difference  columns, 

228.  The  nth  term  of  the  sequence,  in  terms  of  its  first  term 
and  the  first  terms  of  the  first  n  difference  columns. 

Let  the  first  term  in  the  kth  difference  column  be  denoted  by 
A^Wo-     Then  we  have 

Wo  =  Wo, 
AiWo  =  ui  —  Uo, 
A2W0  =  U2  —  2ui+uo, 
A3U0  =  W3  —  3  W2  +  3  wi  —  Uo, 

Solving  successively  ior  uq,  ui,  U2,  .  ■  .  ,  we  have 

Wo  =  Wo, 

wi  =  Wo  +  AlWo, 

W2  =  tio  +  2  AlWo  +  Aotto, 

W3  =  Wo  +  3  AiWo  +  3  A2W0  +  A3?*o, 

Here  the  coefficients  again  follow  the  binomial  law,  and  there  is 
suggested  the  formula 

(1)  W„  =  Wq  +  nCiAiWo  +  nC2^2U0  +    •     '     •     +  A„Wo- 

F' Assuming  the  formula  true  for  m„,  we  can  show  that  it  holds 
for  Un+i.  For  apply  formula  (1)  to  the  nth  term  of  the  first 
order  of  differences,  which  is  Un+i  —  Un.     We  obtain 

Wn  +  l  -  ^ln  =   AlWo  +  nClAzWo  +  „C2A3Wo  +  '  •  "  +  A„  +  iWo. 

Adding  equation  (1)  to  this  we  get 

Wn  +  l  =  Wo  +(„Ci  +  1)  AlWo  +(„C2  +  nC\)   A2W0 

+  (nC3  +  nCa)  A3W0  +  •  •  •  +  A„+iWo. 

But 

„Ci  +  l=n  +  lCi,  „C2  +  «Ci=n  +  iC2,  n^S  +  n^.  =  n  +  l^g,  •  •  •  , 

as  is  easily  verified  by  substituting  in  the  values  of  the  binomial 
coefficients.     Hence 

Wn  +  l=Wo+n  +  lCiAiWo  +  n+lC2A2Wo+n  +  lC3A3Wo+     "    '    "     +A„+iWo. 

Hence,  if  (1)  holds  for  m„,  it  also  holds  when  n  is  replaced  by 
n  +  1,  that  is,  for  Un+\.  But  we  have  shown  that  it  holds  for  W3; 
hence  it  holds  for  u^,  hence  for  wg,  and  so  on. 


229]  DIFFERENCES  205 

229.  The  sum  of  the  first  n  terms  of  the  sequence,  in  terms 
of  its  first  term  and  the  first  terms  of  the  first  7i  —  1  difference 
columns. 

From  the  equations  just  preceding  formula  (1)  we  have,  by- 
addition, 

uo  =  Uq, 
iio  +  wi  =  2^0  +  AiWo, 
Wo  +  wi  +  1^2  =  3  Wo  +  3  AiWo  +  AoUo, 

2^0  +  Ml  +  W2  +  W3  =  4  Wo  +  6  AlWo  +  4  A2W0  +  AsllQ. 

The  coefficients  on  the  right  are  respectively  those  of  the  expan- 
sions of  (1  +  xy,  (1  +  x)-,  (1  +  x)^,  and  (1  +  x)\  the  first  term  of 
the  expansion  being  omitted  in  each  case.  Let  s„  denote  the  sum 
of  the  first  n  terms  of  the  sequence; 

S„  =  Wo+Wi+W2+    •    •    •    +W„_i. 

Then  by  analogy  with  the  preceding  equations  we  assume  that 

(2)  S„  =  „CiWo  +  nCo^ltk)   +  nC3A2Wo  +  „C4A3Wo  H (-  A„_iWo. 

We  show  by  induction  that  (2)  holds  for  all  values  of  n.  Adding 
(1)  of  (228)  to  (2)  and  noting  that  Sn+i  =  s„  +  Un,  we  have 

5n  +  l  =  (.Ci  +  l)Wo  +  UC2  +  „C,)AiWo  +  UC3+„C2)A2Wo+---+A„Wo 
=  „  +  lCiWo  +  n  +  lC2AiWo+,i  +  lC3A2l<0+     '    *    •     +  A^Wfi. 

Therefore  (2)  is  true  when  n  is  replaced  by  w  +  1.  But  we  veri- 
fied above  that  (2)  is  true  when  n  =  4.  Hence  it  is  true  when 
n  =  5,  hence  when  71  =  6,  and  so.  on. 

When  the  rth  order  of  differences  is  zero,  all  following  orders  of 
difference  are  also  zero.  Hence  any  term  of  the  sequence  and  the 
sum  of  any  number  of  terms  can  be  expressed  in  terms  of  the  first 
term  of  the  sequence  and  the  first  terms  of  the  first  r  —  1  difference 
columns.  For  then  formulas  (1)  and  (2)  both  stop  with  the  term 
involving  A^-iWo,  and  we  have 

(3)  w„  =  Wo +  „CiAiWo  +  „C2A2Wo  +  •  •  •  +  „Cr_iA,_iWo. 

(4)  Sn  =nClUo  +  nCsAiWo  +  nCaAgW,,  +     •    •    •     +  „CVA,_iWo. 

Example.  Find  the  sum  of  the  squares  of  n  consecutive  integers  beginning 
with  10. 

Sn  =  102  +  112  +  122  +  .  .  .  +  (10  4-  n  -  1)2. 


206 


INTERPOLATION 


[230 


Our  difference  table  is  as  follows: 

Ao 

Ai 

A2     . 

A3 

100 

21 

121 

23 

2 

0 

144 

25 

2 

0 

169 

27 

2 

196 


Hence  r  =  3.     Then 

Sn  =  nCim  +  nC2AiMo  +  nC3A2Uo 
n  (n 


n  X  100  + 


1-2 

H2n3  +  57n2+541n) 


^X21+^ 


l)(n-2) 


1.2-3 


X2 


Exercises. 

1.  Find  the  sum  of  the  squares  of  the  integers  from  1  to  n  inclusive. 

2.  Find  the  sum  of  the  cubes  of  the  integers  from  1  to  20  inclusive. 

3.  How  many  balls  in  a  square  pyramid  whose  base  has  n  balls  on  a  side. 

4.  As  in  exercise  3  for  a  triangular  pyramid. 

5.  Find  the  sum  of  n  terms  of  the  sequence  a,  a  +  d,  a  +  2  d,  .  .  .     . 

6.  Find  the  10th  term  and  the  (n  +  l)th  term  of  the  sequence  50,  72,  98, 
128,162,  ...     .  Ans.   392;  2  n2+ 2071  +  50. 

230.   Interpolation.  —  Suppose  the  terms  of  the  sequence  Uo,ui, 
U2,  .  .  .  to  be  the  values  of  a  function  /  (.r)  for  a  series  of  equally 

values  of  x.     Thus : 
Wo  =fM, 
ui  =f{xo  -\rh), 
U2  =/(.ro  +  2/i), 

Un  =fi.xo  -hnh). 


Y. 

!/= 

f(x)    ' 

X 

\_^ 

' 

o 

> 

<  > 

'  > 

< 

> 

/X 

These  values  are  shown  graphically  in  the  figure,  as  ordinates  of 
the  curve  y  =  j{x).  From  the  equally  spaced  ordinates  given, 
we  wish  to  calculate  intermediate  ones.  This  is  called  inter- 
polation. 

Replacing  the  w's  in  (1)  of  (228)  by  their  values  above,  we  have 


(5)  /  (rro  +  nh)  =  /  (xo)  +  n^^f  (xo)  + 

n{n-  l)(n-  2) 
"^  1.2.3 


n  (n  —  1) 
1  . 2 

A3/(-io)  + 


Ao/  M 


}^-^ 


^3 


230]  INTERPOLATION  207 

This  formula  has  been  derived  when  n  is  a  positive  integer. 
It  is  also  true  for  fractipnal  values  of  ??,  provided  the  series  on  the 
right  converges.  We  shall  not  stop  for  the  proof,  but  merely 
give  some  simple  applications.  In  practical  cases  the  successive 
differences  Ai/(.To),  A2/(;ro),  .  .  .  become  rapidly  small,  so  that 
first  differences  are  usually  sufficient,  second  differences  are  occa- 
sionally needed,  while  third  and  higher  differences  are  required 
only  in  theory  or  in  the  calculation  of  extensive  tables. 

For  fractional  values  of  n,  formula  (5)  gives  values  of  the  func- 
tion intermediate  to  those  in  the  table.  Thus  when  n  =  2^,  we 
get  /  (xq  -\-2^h),  which  is  the  ordinate  to  the  curve  y  =  f(,x)  falling 
midway  between  the  ordinates  f  (xq -\- 2  h)  and  f{xQ  +  3h). 

Example  1.  Given  the  values  of  log  100,  log  101,  .  .  .  ,  log  109  to  five 
decimal  places,  to  calculate  log  100.7  and  log  107.35. 

Here  /(a;)  =  log  x;  xo  =  100;  h  =  I.  To  calculate  log  100.7  we  put  n  =  .7. 
Our  difference  table  is, 


/(x) 

^i/(x) 

^1  f  ix) 

log  100  =  2.00000 
101  =  2.00432 

+ 

.00432 

428 

-  .00004 

102  =  2.00860 

103  =  2.01284 

424 
421 

* 

4 
3 

104  =2.01703 

416 

5 

105  =  2.02119 

412 

4 

., 

lOG  =  2.02531 

407 

5 

107  =  2.02938 

404 

3 

108  =  2.03342 

401 

3 

109  =  2.03743 

Then 

fixo  +  nh)^ 

=  log  100.7  =  log  100  +  .7  X 

.00432  - 

.7(.7  - 
1  X2 

^  X  .00004  + 

(' 

=  2  +  .00302  +  .00000  = 

2.00302. 

Here  the  second  differences  are  so  small  that  they  can  he  neglected,  and 
our  result  is  that  obtained  by  ordinary  or  linear  interpolation.  Graphically 
this  amounts  to  replacing  the  curve  y  =  f  (x)  by  its  chords. 

To  calculate  log  107.35,  it  is  best  to  consider  log  107  as  the  first  term,  or 
/  (.To),  and  put  n  =  .35.  (We  might  take/(xo)  =  log  100  and  put  n  =  7. 35. J 
We  find 

log  107.35  =  log  107  +  .35 X  .00404-  '^^l'^~^^  X  .00003  +    ■  •  •  =2.03079. 

J  X  ^ 

Here  also  second  differences  arc  negligible. 

All  ordinary  tables  are  constructed  so  that  linear  interpolation  'is  sufficient. 


208  INTERPOLATION  [231 

Example  2.     Given  sin  10°,  sin  15°,".  .  .  ,  sin  45°,  to  calculate  sin  17°  20'. 
The  tabular  numbers  and  their  differences  are  given  below : 

A3/(X) 

-  .0006 


fix) 

sin  10°  =  0. 1736 

15°  =  .2588 

20°  =  .3420 

25°  =  .4226 

Ai/(x) 

+  .0852 
832 
806 

^2  fix) 

-  .0020 
26 
32 

30°  =  .5000 

774 

38 

35°  =  .5736 

736 

44 

40°  =  .6428 
45°  =  .7071 

692 
643 

49 

Herexo  =  10°;  h  =  5°; 

then  17 

°20'  = 

22 
xo  +  Y^h  and ' 

Then 

OO  1  c  I 

sin  17°  20'  =  sin  10°  +  ~  X  .0852  -       y^  ^    '  X  .0020 


22/22 

^^^^^         '  '^"        'X  .0006+  •  •  •    =.2979. 


1X2X3 

Here  the  amount  contributed  by  the  second  difference  is  .0003,  so  that 
linear  interpolation  would  have  been  inaccurate. 

231.   Exercises. 

1.  From  the  table  of  example  1  calculate  log  104.6. 

2.  From  the  table  of  example  2  calculate  sin  12°  30',  sin  27°  30',  and  sin 
36°  15'. 

3.  n  "-^^^^  4.     Altitude. 


10° 
12° 
14° 
16° 
18° 
20° 
22° 
24° 
26° 

Calculate  the  tabular  number  Calculate  the  refraction  for  alti- 

when  n  =  22;  when  n  =  33.6.  '  tudes  14°  40'  and  21°  25'. 


Vn  («  -  1) 

10 

0.0711 

15 

465 

20 

346 

25 

275 

30 

229 

35 

196 

40 

171 

45 

152 

50 

136 

Refraction. 

5'  13" 

.1 

4'  22" 

.5 

3'  45" 

.2 

3'  16" 

.6 

2'  54" 

.0 

2'  35" 

.7 

2'  20" 

.5 

2'  7" 

.6 

1'  56" 

.6 

232]  INTERPOLATION       '  ,   209 

6.      Greenwich                        Moon's  Moon's 

mean  time.  right  ascension.  decUnation. 

h  I  h      m  a 

0          i\-  '  5    14     32.14  18°  47'  37".  7 

2         ,".  7'       5    19     49.41  18°  49'  15".  9 

4  5    25      6.62  18°  50'  20". 6 

6  5    30    23.69?  18°  50'  51". 7 

8  5    35    40.59  18°  50'  49". 4 

10  5    40     57.26  18°  50'  13".  7 
Calculate  the  moon's  right  ascension  and  declination  at  0''  35°"  20'  Green- 
wich mean  time. 

6.    From  a  four-place  table  take  log  310,  log  320,  .  .  .  ,  log  400.     Hence 
calculate  log  317.5. 

232.  Differences  as  a  Check  on  Computed  Values.  —  When  a 
number  of  values  of  a  function  are  calculated  for  equal  intervals  of 
the  argument,  the  differences  should,  ordinarily,  vary  in  a  regular 
manner.  An  irregularity  in  one  of  the  difference  columns  indi- 
cates an  error  in  the  tabular  values,  and  often  enables  the  com- 
puter to  determine  the  amount  of  the  error  and  so  correct  it. 
Example. 

log   70  =  1.8451 
75  =  1.8751 
80  =  1.9030 
85  =  1.9284 
90  =  1.9542 
95  =  1.9777 
100  =  2.0000 
105  =  2.0212 
The  irregularity  in  A2  causes  us  to  examine  Ai ;  here  the  differences  .0254 
and  .0258  are  probably  incorrect,  which  throws  suspicion  on  the  tabular  number 
standing  between  them,  namely  1.9284.     This  number  should  evidently  be 
larger,  and  by  trial  we  find  that  1.9294  is  probably  the  correct  value. 
Exercises.     Correct  the  following  tables: 

'        3. 


Ai 

A2 

.0300 

279 

-  .0021 

254 

25 

258 

4 

235 

23 

223 

12 

212 

11 

15°  = 

.268 

16°  = 

.287 

17°  = 

.306 

18"  = 

.325 

19°  = 

.344 

20°  = 

.369 

21°  = 

.384 

22°  = 

.404 

23°  = 

.425 

24°  = 

.445 

rfi 

2.0 

.250 

2.2 

.207 

2.4 

.174 

2.6 

.158 

2.8 

.127 

3.0 

.111 

3.2 

.098 

3.4 

.087 

3.6 

.077 

Altitude. 

Refraction, 

10° 

5'    13" 

11° 

4'   46" 

12° 

4'   22" 

13° 

4'     2" 

14° 

3'   45" 

15° 

3'   34" 

16° 

3'    16" 

17° 

3'      4" 

18° 

2'    54" 

19° 

2'    35" 

CHAPTER  XV 

Undetermined  Coefficients.     Partial  Fractions 

233.   A  useful  method  for  expanding   certain  expressions  in 
series  depends  on  the  following  Theorem  on  Power  Series. 
If  the  equation 

(1)  ao  +  aix  +  a2X^  +  •   •   •   +  a„a;"  -f  •   •   •  =  0 

is  true  for  all  values  of  x  from  a:  =  0  to  a;  =  a:o  inclusive,  where 
Xq  ^  0,  then  all  the  coefficients  are  zero,  that  is, 

ao  =  0,  ai  =  0,  a2  =  0,  .  .  .  ,  a„  =  0,  .  .  .    . 

Proof.     Since  (1)  is  true  when  a;  =  0  we  have,  putting  0  for  x, 
ao  =  0. 

Then  (1)  reduces  to 

aix  +  a2X^  +  •   •   •   +  a^a;"  +  •  •  •    =0, 
or 

(2)  a;(ai+a2a:+  •  •  •  +a"x"-i+   •  •  •  )  =  0. 

This  must  be  true  for  all  values  of  x  from  0  to  a;o.  Choose  for  x  a 
value  £  between  0  and  a;o.     Then 

£(ai+a2s+   •  •  •  +a''£"-i+   .  .  .  )=0. 

Then,  since  £  5^  0,  we  must  have 

ai+a2£+  •  •  •  +a,j£''-i+   .  .  .   =  0, 
or, 

ai  =  -  £  (as  +  age  +  •  •  •  +  a„£"-2  +...). 

The  series  in  the  last  parenthesis  converges,  and  therefore  has 
a  finite  sum  S.  For,  putting  a:  =  £  in  (1),  and  omitting  the  first 
two  terms,  we  have  left  the  convergent  series 

a^e'-  +  a3£3  +   .  .   .  +  a,,s^  +   •   •  •  , 
and  this  remains  convergent  after  division  by  £2.     Hence 
ai  =  —  sS 

where  S  depends  on  e,  but  is  finite  for  all  values  of  £  between  0 

210 


234]  UNDETERMINED  COEFFICIENTS  211 

and  xq.  Assume  now  that  ui  is  not  equal  to  0;  say  ai  =  h.  We 
can  now  take  s  so  small  that  eS  shall  be  numerically  less  than  h; 
hence  ai  cannot  equal  h.     .'.   ai  =  0. 

Then  (1)  reduces  to 

aox-  +  a-sx^  +   •   •   •  +  anX""  +   •   •   •    =  0, 
or,  x^  (a2  +  asa;  +   •  •  ■  +  anx"-~  +...)=  «• 

Choose  for  x  a  value  e  (not  necessarily  the  same  as  e  above)  between 
0  and  Xq.     Then 

£2(«2  +  a3^'+  •  •  •  +an£"-2+   .  .  .  )=0. 
Hence,  since  e  9^  0,  we  have 

02  +  «3=-  +   •   ■   •  +  ans"""  +   •  •   •    =  0, 
or,  02  =  -  c  (03  +   •   •   •  +  a„e--'  +•••)=  0. 

Here  again  the  series  in  parentheses  converges  and  has  a  finite 
sum.  Hence  by  taking  e  sufficiently  small  we  can  show  that  02 
cannot  equal  any  number  h,  however  small.      /.  02  =  0. 

Similarly  we  show  that  each  coefficient  must  be  zero. 

234.  Theorem  of  Undetermined  Coefficients.  —  If  two  power 
series  in  x  are  equal  to  each  other  for  all  values  of  x  from  a:  =  0  to 
X  =  xo  inclusive,  then  the  coefficients  of  like  powers  of  x  in  the 
two  series  must  be  equal. 

Hypothesis: 

(1)    ao  +  aix  +  a2X"  +  ■  •  •  +  anX"  +  •  .  •   = 

60  +  bix  +  &2a:2+  .  .  .  -f  6„a;"  +  •  •  •  when  0  ^  a;  ^  a^o- 

Conclusion: 

ao  =  bo,  ai  =  61,  a2  =  &2,  •  •  •  ,  ct/i  =  &n,  •  •  •    • 
Proof.     From  (1),  by  transposition,  we  have 
ao-&o  +  (ai-&i)^+(«2-&2)a;2+  •  •  •  +(a„-6„)a:"+  •  •  •  =  0. 
Hence  by  the  preceding  theorem, 

ao  -  60  =  0,  ai  -  6i  =  0,  a2  -  62  =  0,  .  .  .  ,  a„  -  h„  =  0. 

Hence  the  conclusion  stated  above. 

Corollary.  The  theorem  remains  true  when  either  or  both 
of  the  infinite  series  reduce  to  polynomials.  We  consider  a  poly- 
nomial of  7n  terms  as  an  infinite  series  in  which  all  coefficients 
after  the  mth  are  zero. 


212 


UNDETERMINED  COEFFICIENTS 


[234 


Assume 


1  +x 


1    -X2 
+  X  - 

=  ao  +  aix  +  02x2  +  asx^  + 


Y    X'' 

Example  1.     Develop  t— : — ^^—^  into  a  power  series. 
1  - 


Clearing,  and  writing  the  coeflBcients  of  like  powers  of  x  in  vertical  columns,  we 
have 


1  -x2 


+  ai 


X  +  a2 

x2  +  as 

+  ai 

+  a2 

-  ao 

-ai 

X3  + 


Equating  coefficients  of  like  powers  of  x,  we  have 

ao  =  1,  or,       oo  =  1, 

oi  +  oo  =  0,  ai  =  —  1, 

02  +  ai  —  ao  =  —  1,  02  =  1, 

OS  +  02  —  oi  =  0,  as  =  —  2. 

Hence 

1  - 


1  +x 


=   1-X  +  X2-2X3+    • 

1  +  2  X 
Example  2.     Develop  ^ ,    »    .    ^3  into  a  power 

D  X  OX     ~\~  X 


If  we  put 


1  +2x 


i  X   -  5  X2  +  X3 


Oo  4"  OlX  +  a2x2  + 


clear  of  fractions  and  equate  coefficients,  we  have  to  begin  with  1=0.  This 
absurdity  results  from  the  fact  that  we  have  not  taken  a  proper  form  for  the 
development.     By  inspection  we  see  that  the  quotient  of  1  +  2  x  divided  by 

6  X  —  5  x2  +  x3  should  start  with  — .     To  obtain  the  development  we  put 


1  +2x 


1  1  +2x 


6x-5x2+x3      X     6-5xH-x2 
Developing  the  last  fraction  as  in  example  1, 
1  +2x 


Hence 


_lj_17      4_Ii     24_.293     3 

6-5x  +  x2~6  +  36^^216''    "^1296^    "^ 

1+2X         _    1        17       79        .    j93^ 
)  X  -  5  x2  +  x3      6  X  ^  36  ^  216     ^  1296      ^ 


t 


Exercises. 

1.   In  example  1,  find  an  in  terms  of  on -1  and  on-2. 
■ind  the  first  four  terms  of  the  expansions  of: 

1+  X  ,  X  ^  1   +  X2 


1  +X  +X2" 

1    -X 
1    -  X   -X2' 


2  -  X  +  3  x2 

2  x^  +  3  X 
x2  +  2x  +  2' 


1  +  3  X  +  a;3 

2  -  3  X  +  x2 
3x  +  4x2  -x3' 


235] 


PARTIAL  FRACTIONS  213 


235.  Partial  Fractions.  —  It  is  sometimes  desirable  to  resolve  a 
given  rational  fraction  into  a  sum  of  simpler  fractions,  called  'par- 
tial fractions.  This  can  be  done  when  the  denominator  of  the 
given  fraction  can  be  factored.  Several  cases  arise,  according  to 
the  nature  of  these  factors. 

For  reasons  which  will  presently  appear,  the  methods  to  be  ex- 
plained apply  only  to  fractions  in  which  the  degree  of  the  numerator 
is  less  than  the  degree  of  the  denominator.  When  this  is  not  the  case, 
divide  numerator  by  denominator  until  a  remainder  of  less  degree 
than  the  denominator  is  obtained. 

Case  1.  The  denominator  can  be  factored  into  linear  factors 
of  the  form  {ax  +  6),  no  two  factors  being  equal. 

Rule.  The  fraction  can  be  resolved  into  a  sum  of  simple  frac- 
tions, of  the  form  — r ,  equal  in  number  to  the  factors  of  the 

'  ax  -{-0 

given  denominator.     Here  A  is  a  constant. 

5a; -1  5x-l ■    .     A       .      B 

Example.       ^2  _  6x  +  5  "  (x  -  l)(x  -  5)  "  x  -  1  +  x  -  5* 

Clearing:  5  x  -  1  =  A  (x  -  5)+ 5  (x  -  1), 

or,  5x-l  ={A+B)x -{5 A +B). 

Since  the  given  fraction  must  be  equal  to  its  partial  fractions  for  all  values 

of  X  except  x  =  1  and  x  =  5,  the  last  equation  must  be  true  for  all  such  values 

of  x;  hence  we  equate  coefficients  of  like  powers  of  x  (233,  Corollary).     We 

obtain 

5  =  A+B;     -l=-i5A+B). 

Hence  A=-l;  5=6. 

5x-l       ^    -1     .       6 

x2-6x  +  5      x-l^x-5' 

A  shorter  method  for  finding  A  and  B  is  as  follows:  consider  again  the 

equation 

5x-l=Aix-5)+B{x-l). 

Let  x=5;        24  =  4  5;  B  =  6. 

Let  x  =  l;  4  =-4  J.;        A=-\. 

We  can  justify  the  use  of  the  values  x  =  1  and  x  =  5,  for  which  the  given 
fraction  and  one  of  the  partial  fractions  become  infinite.     For  the  equation 
5x-l  ^     _1_     ^ 


x2-6x  +  5      x-l'x-5 
must  hold  except  when  x  =  1  or  x  =  5. 

Hence 

5x-l  =A(x-5)+5(x-l) 


214  PARTIAL  FRACTIONS  [236 

is  true  for  all  values  of  x,  except  perhaps  x  =  1  and  x  =  5.     It  is  therefore 
true  when  x  =  1  +  £,  however  small  e  may  be  ;  that  is, 

(1)  5(l  +  £)-l    =^(l  +  £-5)+5(l+£-l). 

Suppose  our  equation  is  not  true  when  x  =  1 ;  let  the  two  members  differ  by 
a  quantity  h,  so  that 

5X1-1  =A(l-5)+B(l-l)+/i, 
or,  4=-4A+/i. 

From  (1)  we  have 

4  +  £  =-4^  +  £A  +  £fi. 

From  the  last  two  equations,  by  subtraction,  etc., 

Since  A  and  B  are  fixed  numbers,  h  can  be  made  as  small  as  we  wish  by  taking 
£  small  enough.     Hence  h  cannot  equal  any  number  except  0.    . 

236.  Case  2. — The  denominator  contains  a  linear  factor  repeated 
r  times,  as  {ax  +  hy. 

Rule.  Corresponding  to  the  factor  {ax  +  hf,  take  a  set  of  par- 
tial fractions  of  the  form 

Ai  A2  _._  Ar 

{ax  +  6)  ^  {ax  +  6)2  "^  '         "^  {ax  +  6)'- 
This  is  the  most  general  set  of  fractions  having  constant  numer- 
ators and  common  denominator  {ax  +  hy. 

Example. 

3  x2  -  X  + 1  A  B  C  D 

(X  +  2)(x  -  3)3      X  +  2  "^  X  -  3  "^  (X  -  3)2  "^  (X  -  3)3' 
Clearing : 

3  x2  -  x  +  1  =  A  (x  -  3)3  +  B  (x  +  2)(x  -  3)2  +  C  (x  +  2)(x  -  3)  +  D  (X  +  2). 
Let  X  =  3;        then     25  =  5  D;  D  =  5. 

Let  X  =  -  2;     then     15  =  -  125  ^;        A  =-  is- 

Since  no  other  factors  are  available  to  furnish  other  values  of  x  for  substitu- 
tion, we  choose  any  convenient  values,  say  x  =  0  and  x  =  1. 
Put  x=0;     1=-27A  +  18B-6C  +  2D. 

Put  x  =  l;     3  =-    8^  +  12B-6C  +  3D. 

Substituting  the  values  of  A  and  D  already  found,  and  solving  for  B  and  C, 
we  have 

Hence 

3x2-x  +  l     _       -3  3 ■  ^_12_^  .   _J__. 

(x  +  2)(x-3)3  ~  25  (X  +  2)  "^  25  (x  -  3)  "^  5  (X  -  3)2  "^  (x  -  3)3 


237]  PARTIAL  FRACTIONS  215 

237.  Case  3.  —  The  denominator  contains  a  quadratic  factor, 
{ax~  +  bx  4-  c),  which  cannot  be  resolved  into  real  linear  factors. 

Rule.  Corresponding  to  a  quadratic  factor  {ax^  -^bx  -{-  c),  take 
a  partial  fraction  of  the  form 

Ax  +  B 
ax^  +  6x  +  c 

The  reason  for  this  assumption  may  be  illustrated  by  a  simple 
example. 

2  X  —  1 
Example.     Resolve       _  -iw  2  -|-4>  ^^*°  partial  fractions. 

If  i  =  V  —  1.  the  factors  of  x2  +  4  are  x  +  2  i  and  x  —  2i.  Suppose  now 
we  assume 

2x-  1  A  B  C 

~i    ~    I    o  V  "I"  ! 


(x-l)(x2  +  4)       x-1    '   x  +  2i   '   x-2i 

Combining  the  last  two  fractions  into  a  single  one,  we  have 

B  C      ^  {B  +  C)x  +  2{C-B)i 

x  +  2i      x-2i  x2  +  4 

If  now  we  introduce  two  new  constants  M,  N  in  place  of  B,  C,  by  the  relations 

B  =  M  +  iN;     C  =  M  -  iN, 
we  have 

B  +  C  =  2M;    i{C  -B)  =  -2v^N=2N. 

Hence  in  place  of  the  fractions 

B       ,        C 

x  +  2i      x-2i' 

where  B  and  C  involve  i,  we  take  the  single  fraction 

Mx+4Ar 
x2  +  4     ' 

where  M  and  N  are  real.     Then,  using  B  in  place  of  M  and  C  in  place  of  4  N, 
let 

2x-l  A         Bx  +  C 

(x  -  l)(x2  +4)      X  -  1  "'"  x2  +  4  ■ 

Clearing:  2  x  -  1  =  A(x2  +  4)  +  (Bx  +  C)(x  -1). 

Put  x  =  l;     then        1=5  A;  A  =  I. 

Put  X  =  0;     then    -  1  =  4  A  -  C;  C  =  |. 

Equate  coefficients  of  x*;       then     0  =  A  +  B;  B  =-  A  =-|. 

Hence 

2x  -  1  1  ,     -  X  +  9 


(X  -  1)  (x2  +  4)      5  (x  -  1)    '  5  (x2  +  4) 


216  PARTIAL  FRACTIONS  [238,239 

238.  Case  4.  —  The  denominator  contains  a  repeated  quadratic 
factor,  {ax^  -\- hx  -}-  cY. 

Rule.  Corresponding  to  a  repeated  quadratic  factor  {ax^  + 
6a;  +  cY,  take  the  partial  fractions, 


+  7-i^fTe^Tv>+    •    •    •    + 


{ax^  +  bx  +  c)'^  (aa;2  -{-bx  +  c)^^  '   {ax'  -}-bx  +  cY 

Example. 

10  x3  +  7  X  +  4  A  Bx  +  C       Dx  +  E 

(X  -  2)  (x2  +  3)2       X  -  2  "^  x2  +  3    "^  (x2  +  3)2 
Clearing: 

10x3  +  7x  ,+  4  =  A  (x2  +  3)2  +  (Sx  +  C)  (x  -  2)  (x2  +3)+  (Dx  +  E){x  -  2). 
Put  X  =  2;     98  =  49  A;    A  =  2. 

Equate  coeflBcients  of  x^,  x^,  x2,  and  x": 

0  =  A  +  5, 
10  =  C  -  2  5, 
0  =  6.4+3B-2C  +  D, 
4  =  9A-6C-2J5;. 
Hence,  B  =  -  2,   C  =  6,   D  =  6,   £;  =  -  11. 

Therefore, 

10  x3  +  7  X  +  4  2  -2x  +  6      6x  -  11 

(X  -  2)  (x2  +  3)2      X  -  2  ^     x2  +  3     "^  (x2  +  3)2 

239.   Exercises.     Resolve  into  partial  fractions: 

-^- 

x6  +  x"  -  8 
x3  -  4x 

5x  +  12 


3  x2  +  10  X  +  3 

3x  -  1 

8 

x2  +  X  -  6 

x2  +  6  X  -  8 
x3    -  4x 

9. 

1+X2 

10. 

X   -X3 

X 

11. 

x2-4x  +  l 

X4 

12. 

x3  +  2x2-x  -2 

in         3x2  -2x 

■•    x3  +  4  X 

,      1   . 

'•     (X2-1)2 

X3   -  1 

'•    X3  +  3  X 

,          .3  +  1 

.^.  •       20.^'  +  ^^+-^ 


X  (X  -  1)3 
x3  -  3  X2  +  2  X 

x2  +  3  X  +  4 

x3  +  2  x2  +  x'  '""•  x4  +  3  x2  +  2 


13. 

X  -  8 
x3  -  4  x2  +  4  X 

8 

14. 

1 

X4  +  x3  +  X2   +  X 

15. 

1 

X3  +   1 

16. 

X2   -   1 

x2  -  4 

17. 

x2  -3 

x3  -  7  X  +  6 

18. 

x5  -  2  X  +  1 

X"  +  2  X3  +  X2 

21. 

X2 

+  8 

x  +  4 

X3  + 

X2   - 

-4x  -4 

9.?.. 

X2  - 

■  2x 

-  1 

CHAPTER  XVI 


Determinants 

240.  Determinants  of  the  Second  Order. 

taneous  linear  equations 

aiX-\-biy  =  ci, 
aox  +  62^  =  C2, 

are  solved  for  x  and  y,  we  find 


When  two  simul- 


62C1  —  b\C2 


y  = 


aiCo  —  (I2C1 


aib2  —  a2bi  '  aibo  —  a2&i 

To  express  these  results  it  is  convenient  to  use  the  notation 

lai  fei 


02  fe^ 


(0162  —  «2fcl), 


where  the  square  array  between  vertical  bars  is  simply  another 
way  of  writing  the  expression  forming  the  right  member  of  the 
equation.  It  is  called  a 'determinant,  and  in  particular,  a  deter- 
minant of  the  second  order,  because  there  are  two  rows  and  two 
columns.  The  quantities  ai,  61,  02,  62)  are  called  the  elements  of 
the  determinant. 

The  value  of  a  determinant  of  the  second  order  may  be  obtained 
by  forming  the  products  of  elements  which  constitute  the  diagonals 
of  the  array  and  giving  these  products  the  signs  indicated  in  the 
scheme  below: 


M' 


This  process  is  called  "  expanding  the  determinant." 

The  above  values  of  x  and  y  may  now  be  written  in  the  forms, 


c\  bi 

a  I   C] 

X  = 

Co   bo 

'  y  = 

02  C2 

ai   61 

ai   61 

a2  62 

a2   &2 

217 


218 


DETERMINANTS 


[241 


Exercises. 

1.  State  a  rule  for  writing  the  above  values  of  x  and  y. 
Solve  for  x  and  y,  by  aid  of  determinants: 

2.  X  -  y  =  I,  3.   4x-32/  =  5,  4.   8x  +  5y-6  =  0, 
2x  +  ?/  =  3.                     2x  +  ?/  =  l.  4x  +  y  +  4=0. 

5.   2x+?/  +  l=0,  6.   2x  +  y  +  l=0, 

6x  +  3y  +  2=0.  6x  +  32/  +  3=0. 

241.  Determinants  of  the  Third  Order.  —  We  shall  now  define 
a  determinant  of  the  third  order  in  terms  of  determinants  of  the 
second  order  by  the  following  equation: 


fli  a2  as 

fli 

62    &3 

-  02 

61    63 

+  ^3 

&l    &2 

&1    62    &3 

— 

C2     C3 

Cl     C3 

Cl     C2 

C\     Co     C3 

where  the  determinants  on  the  right  are  to  be  expanded  and  the 
results  multiplied  by  the  quantity  written  in  front  of  the  determi- 
nants respectively. 

On  performing  these  operations  and  collecting  terms,  we  have 


a3&2Cl  —  a2&lC3  —  ai&3C2- 


h  \^^^^  \  aih2C3  +  aobsCi  +  036102 
Cl  \\\j 

This  is  the  expanded  form  of  a  determinant  of  the  third  order, 
and  may  be  written  out  by  forming  the  products  of  the  terms 
joined  by  arrows  in  the  scheme  below,  each  product  to  be  given 
the  sign  indicated. 


+^ 


We  may  now  verify  by  direct  calculation  that  the  values  of  x, 
y,  z,  obtained  by  solving  the  linear  equations 
aix -\- hiy -\- ciz  =  di,  ' 
a2X  +  62?/  +  C2Z  =  ^2, 
asx  +  h^y  +  c^z  =  dz, 


242] 


DETERMINANTS 


219 


di  bi  ci 

ai  di  ci 

ai  6i  di 

d2  bo  c-i 

ao  do  C2 

a2   62  4 

ds   b;i   Cs 

'  y  = 

as  d^i   C3 

'  z  = 

«3  ^3  ^^3 

ai  61  ci 

ai   61  ci 

ai  61  Ci 

02  &2  C2 

a2   62  C2 

02  62  C2 

^3  h   C3 

aa  &3  C3 

as  63  C3 

X  +  2  2/  -It,3  2  =  2, 

3x  +  2z/+     2  =  3. 

^.   2x  +  2y  -     2  =  2, 

x+      2/ -22  =  1, 

X  -     y  +     2  =  4. 


X  -  ?/  +  2  =  2, 
2x+  y  +  3z  =  l, 
2x-2y  +  2z  =  4. 
2x-     y  +  2z=2, 

x-2?/  +  42=3, 
3x  -3?/  +62  =  1. 


Exercises. 

1.  Verify  the  last  statement. 

2.  State  a  rule  for  solving  three  equations  of  the  form  just  considered. 
Solve  the  following  systems  of  equations: 

X-     y+    z  =  \,       5.   5x  +  62/-32  =  4,         7. 
4x-5?/  +  22  =  3, 
2x-Zy+     2  =  1. 
5.   3x-6j/  +  9z  =  2,        8. 
x+     y+     2  =  1, 
x-2;/  +  32  =  2. 

9.  Show  that  a  determinant  of  second  or  third  order  vanishes  when  the 
elements  of  a  row  or  column  are  equal  respectively  to  those  of  another  row  or 
column. 

10.  Show  that  a  determinant  changes  sign  when  the  signs  of  all  the  elements  ' 
of  any  row  or  column  are  changed. 

11.  Show  that,  if  the  elements  of  any  row  or  column  be  multiplied  by  a 
factor  k,  the  determinant  is  multiplied  by  k. 

242.   Inconsistent  or  Non-independent   Linear   Equations. — 

Consider  the  equations 

aix  +  bxy  =  Ci     and     kaix  +  kbiy  =  Co. 

These  are  inconsistent  if  C2  7^  kci;  they  are  dependent  if  Co  =  kci, 
since  in  this  case  the  second  equation  is  k  times  the  first. 

In  either   case   the  determinant  of  the  coefficients  of  x  and  y 
is  0.     On  solving  by  the  determinant  method,  we  find 

X  =  00  and  y  =<x> ,  when  the  equations  are  inconsistent; 


a;  =  ^    and 


y 


when  the  equations  are  dependent. 


That  is,  the  inconsistent  equations  have  no  (finite)  solution,  while 
the  solution  is  indeterminate  in  case  of  dependent  equations. 

Geometrically,  the  equations  represent  two  straight  lines  which 
are  parallel,  and  distinct  if  C2  7^  c\k;  they  coincide  if  c^  =  cik. 


220 


DETERMINANTS 


[243 


Hence  the  infinite  values  of  x  and  y  above  are  equivalent  to  the 
statement,  "  Parallel  lines  meet  at  infinity."  In  the  second  case, 
when  the  lines  coincide,  the  coordinates  of  any  point  on  either 
line  satisfy  both  equations.  Hence  there  are  an  infinite  number  of 
solutions,  and  hence  x  and  y  appear  above  as  indeterminate  forms. 
[See  exercises  5  and  6  of  (240).] 

Exercises.     1.    Consider  the  equations 
'  aix  +  h\y  +  ciz  =  di,    kaix  +  khiy  +  kciz  =  6,2,    asx  +  bsy  +  czz  =  ds. 

The  first  two  are  inconsistent  if  ^2  9^  kd\,  and  dependent  when  di  =  kdi. 
Show  that  in  the  first  case  the  only  possible  sokitions  of  the  three  equations 
are  infinite,  and  in  the  second  case  there  is  an  infinite  number  of  solutions. 

2.  Show  that  the  equations 

aix  +  h\y  =  0     and     a2X  +  h^y  =  0 

have  one  solution  (0,  0),  or  an  infinite  number  of  solutions,  according  as  the 
determinant  of  the  coefficients  is  different  from  or  equal  to  0.  Discuss  also 
geometrically. 

3.  Show  that  the  equations 

aix  +  h\y  +  c\z  =  0,     a2X  +  62?/  +  C2Z  =  0,     asx  +  hy  +  c^z  =  0 

have  one  solution  (0,  0,  0),  or  an  infinite  number  of  solutions,  according  as  the 
determinant  of  the  coefficients  is  different  from  or  equal  to  zero. 

{Hint.     Eliminate  z  so  as  to  get  two  equations  in  x  and  y  and  discuss  these 
as  in  exercise  2.) 

4.  Show  that  the  equations 

2x-32/  +  52  =  0,     x  +  y-2=0,     3x-7?/  +  llz  =  0 

are  not  independent.     What  is  the  relation  between  them? 

{Hint.     To  find  the  relation  between  the  equations,  find  ^1  and  k2  such  that 
ki  times  the  first  trinomial  plus  /c2  times  the  second  shall  equal  the  third.) 

243.   General  Definition  of  a  Determinant.  —  The  array  of  n 
rows  and  n  columns, 


ai  a2  as  . 

.  a, 

61    62    &3    • 

.  6, 

Cl      C2    Cg    . 

.  .  c, 

l\     h    h  ■  ■  • 

is  called  a  determinant  of  order  w.     The  quantities  forming  the 
array  are  called  the  elements  of  the  determinant. 


244,245]  DETERMINANTS  221 

If  we  form  all  possible  products  of  n  elements,  each  product  to 
contain  one  and  only  one  element  from  each  row  and  column, 
and  if  these  products  are  given  proper  signs,  as  will  presently  be 
indicated,  and  added  algebraically,  the  sum  so  obtained  is  defined 
to  be  the  value  of  the  determinant. 

Each  product  of  n  elements  so  obtained  is  called  a  term  of  the 
expanded  form  of  the  determinant. 

The  elements  ai,  b^,  C3,  .  .  .  ,  l,,  form  the  principal  diagonal. 

The  term  aihoCs  .  .  .In  is  called  the  principal  term  of  the  ex- 
pansion. 

244.  Every  term  of  the  expansion  of  the  determinant  can  he 
formed  from  the  principal  term  by  rearranging  the  subscripts,  leav- 
ing the  letters  in  their  natural  order. 

For  every  term  contains  all  the  letters  and  all  the  subscripts, 
and  each  only  once,  since  it  is  a  product  containing  one  and  only 
one  element  from  each  row  and  each  column.  Hence  if  the  letters 
in  any  term  be  arranged  in  their  natural  order,  the  subscripts  will 
form  some  arrangement  o^the  numbers  1,  2,  3,  .  .  .  ,  n. 

Conversely,  every  rearrangement  of  subscripts  in  the  principal 
term,  the  letters  being  left  in  their  natural  order,  yields  a  term  of 
the  expansion,  since  it  contains  one  element  and  onl}'  one  from 
each  row  and  each  column. 

Therefore  all  the  terms  of  the  expansion  can  be  obtained  by 
forming  all  possible  arrangements  of  subscripts  in  the  principal 
term. 

We  shall  use  the  symbol  An  to  indicate  our  determinant  of 
order  n.     Then  we  can  write  the  equation 

A„  =  S  ±ai&2C3  ■  ■  '  In,      (^  =  sigma) 

where  the  symbol  S  (sign  for  a  sum)  means  that  we  are  to  form 
the  algebraic  sum  of  all  terms  which  may  be  formed  from  the  term 
written  by  forming  all  possible  arrangements  of  the  subscripts; 
the  signs  of  the  terms  so  formed  remain  to  be  determined. 

245.  Number  of  Terms  in  the  Expansion  of  A„.  —  The  num- 
ber of  terms  in  the  expansion  of  a  determinant  of  order  n  is 
1  X  2  X  3  X   •  •  •   X  n,     or     |^- 

Proof.  We  need  only  show  that  the  number  of  possible  arrange- 
ments of  the  subscripts  1,  2,  3,  .  .  .  n,  is  jw. 


222  DETERMINANTS  [246,247 

Starting  with  the  natural  order,  and  interchanging  1  in  turn 
with  2,  3,  .  .  .  ,  n,  we  form  the  n  arrangements 

1  2  3  ...  n, 

2  1  3  ...  w, 
2  3  1  ...  n, 

2  3  4  ...  1. 
In  any  one  of  these,  keep  1  fixed  in  its  position,  and  interchange  2 
with  3,  4,  .  .  .  ,  n.  In  this  way  we  form  n  —  \  arrangements  in 
which  1  occupies  a  given  place.  Treating  each  of  the  n  arrange- 
ments written  above  similarly,  we  obtain  altogether  n  (n  —  1) 
arrangements.  Each  of  these  gives  rise  to  a  group  of  n  —  2  ar- 
rangements, including  itself,  by  interchanging  3  with  4,  5,  .  .  . ',  n. 
Hence  we  obtain  n{n—l){n  —  2)  arrangements.  Proceeding  simi- 
larly we  find  the  total  number  of  arrangements  to  be  |n. 

246.   Signs  of  the  Terms  in  the  Expansion  of  A„. 

Inversion.  An  arrangement  of  the  numbers  1,  2,  3,  .  .  .  ,  n 
is  called  an  inversion.  An  inversion  is  even  or  odd  according  as 
the  number  of  times  a  greater  number  precedes  a  lesser  number 
is  even  or  odd. 

Thus,  the  possible  inversions  of  3  numbers  are 
123,  213,  231,  321,  312,  132; 
of  these  the  first,  third,  and  fifth  are  even,  the  others  odd. 

Further,  the  inversion  of  the  subscripts  in  the  term  aj})2Czdi  is 
even.  For  4  precedes  2,  3,  and  1,  and  3  precedes  1,  making  a 
greater  subscript  precede  a  lesser  one  4  times. 

We  now  define  the  sign  of  each  term  of  the  expansion  of  A„  hy  the 
rule  that  the  sign  shall  he  plus  when  the  inversion  of  the  subscripts  is 
even,  minus  when  the  inversion  is  odd. 

Our  determinant  is  now  completely  defined. 

Exercise.     Write  out  the  expansion  of 

fll    02    03   Cl4 

bi    62  bs  bi 


Ai  = 


Cl     C2    C3    Ci 

di  do  dz  d\ 


247.   Properties  of  Determinants. 

1.   A  determinant  is  unchanged  in  value  when  its  rows  and  col- 
umns are  interchanged. 


247] 


DETERMINANTS 


223 


For  the  expansion  remains  unaltered. 

2.  Interchanging  two  adjacent  rows  or  columns  changes  the  sign 
of  the  determinant. 

For  each  term  of  the  expansion  will  change  sign,  since  two 
adjacent  subscripts  will  be  interchanged;  hence  even  inversions 
change  to  odd,  and  vice  versa. 

By  repeated  application  of  this  rule  it  follows  that  if  any  two 
rows  or  any  two  columns  he  interchanged,  the  sign  of  the  determinant 
changes. 

3.  If  all  the  elements  of  a  row  or  column  are  0,  the  determinant  =  0. 
For  each  term  of  the  expansion  contains  a  zero  factor. 

4.  When  all  the  elements  of  a  row,  or  column,  contain  a  common 
factor,  this  may  he  taken  out  and  written  as  a  factor  of  the  whole 
determinant. 

For  each  term  of  the  expansion  will  contain  this  factor. 
It  follows  that,  to  multiply  a  determinant  hy  any  factor,  we  need 
only  multiyly  the  elements  of  any  row  or  column  hy  this  factor. 

5.  If  two  rows  or  columns  are  alike,  the  determinant  =  0. 

For  by  interchanging  them  we  would  have  A«  =  —  An;  .'.  An  =  0. 

6.  If  the  elements  of  two  rows  or  columns  differ  only  hy  a  common 
factor,  the  determinant  =  0. 

For  by  taking  out  the  common  factor  the  two  rows  or  columns 
become  equal. 

7.  //  in  the  expansion  of  An  we  collect  the  terms  which  contain  the 
several  elements  of  any  row  or  column,  say  the  jth  row,  we  have 


A„  = 


ai  a2  as  .  .  .  an 
hi  b2  hs  .  .  .  fe„ 

il     J2    J3     ■    ■     ■      jn 


U    h    h 


3\Ji  -h  J2J2  +    •   •   •  jnJn 


Here  Ji  is  called  the  cof actor  of  the  element  ji,  and  similarly  for 

/2,    .    .    .    ,    Jn. 

8.  A  determinant  is  unaltered  in  value  when  the  elements  of  any 
row  are  increased  hy  a  constant  multiple  of  the  corresponding  elements 
of  another  row.     Similarly  for  columns. 


224 


DETERMINANTS 


[247 


For  suppose  that  we  add  to  the  elements  of  the  first  row  k  times 
the  elements  of  the  second.     We  obtain  the  determinant 


An'  = 

ai  +  kb 
h 

Cl 

I,  a2-\-kb2,  .  .  .  ,  ttn  +  kb 
62        .  .  .           b^ 

C2           ...               C„ 

h 

I2    .         .    .    .                 In 

AuA2,  . 
,  so  that 

.  .  ,A„ 

be  the  cofactors  of  the  elen 

An  =  (ai  +  A;6i)  Ai+  (a2  +  kb.)  A2  +  •  •  •  +  (a„  +  kbn)  A„ 
=  {aiAi  +  a2^2  +  •  •  •   +  anAn)  + 

A;  (61^1  +  62^2+   •  •  •  +Mn). 


ai  a2  . 

.    Ctn 

61  62  . 

.    &n 

+  k 

h    h    . 

.    .     Zn 

bi  62 
fei  62 

h   h 


The  first  of  these  determinants  is  A„,  the  second  equals  0. 

An'  =  An. 

It  follows  that  we  can  add  to  the  elements  of  any  row  any  linear 
combination  of  corresponding  elements  of  other  rows. 

Example.     Without  expanding,  show  that 

102  104  106 
99  98  97  =0. 
12   3 

Subtract  the  second  row  from  the  first.     The  new  form  is 

3  6  9 

99  98  97 

1  2  3 

This  is  zero,  by  6. 

9.   //  the  cofactors  of  any  row  or  column  be  multiplied  by  the  ele- 
ments of  any  other  row  or  column,  the  sum  of  the  products  is  zero. 


247] 


DETERMINANTS 


225 


For  we  have 


aiAi  +  02^2  + 


+  anAn. 


Replace  the  a's  by  the  elements  of  any  other  row,  as  the  second. 
The  result  is 


biAi  +  62A2  + 


+  Mn  =  0. 


10.  If  we  strike  out  from  A„  the  jth  row  and  kth  column,  the  remain- 
ing determinant,  of  order  n  —  1,  is  designated  by  A/,fc,  and  is  called 
the  minor  of  the  element  standing  at  the  intersection  of  the  row  and 
the  column  struck  out. 

Thus  the  minors  of  ai,  02,  and  03  in  the  determinant 


are,  respectively, 


ai 

ao 

as 

61 

&2 

&3 

Cl 

C2 

C3 

&2  h 

61 63 

,     and 

61  62 

C2    C3 

Cl    C3 

Cl    C2 

m,  the  minor  of  any 
We  shall  consider  a 


We  shall  now  show  that,  except  as  to 
element  equals  the  cofactor  of  that  element 
determinant  of  third  order,  although  the  argument  will  apply  to 
determinants  of  any  order.     We  have 


aiAi  +  a2A2  +  asAs, 


\ai  02  as 
A3  =61   62  63 

Ici     Co     Cs 

where  ^1,  A2,  A3  are  the  cofactors  of  ai,  02,  a^,  respectively. 

Then  Ai  =  Ai.i. 

For,  since  ai^i  contains  all  the  terms  of  A3  which  involve  ai,  and 
since  the  expansion  of  Ai,i  contains  all  possible  products  of  ele- 


226  DETERMINANTS  [  248 

ments,  one  from  each  column  and  each  row  except  the  first,  there- 
fore Ai  and  Ai,i  must  be  identical.  Now  interchange  the  first 
two  columns,  so  that  A3  becomes  —A3.     Then 


ao 

ai 

as 

62 

61 

h 

C2 

Ci 

C3 

A3  =    62    61     63    =  —  02^2  —  fll^l  —  «3-43. 

The  minor  of  a2  is  unchanged,  namely  .     The  expansion 

Cl     C3 

of  this  multiplied  by  a2  gives  all  the  terms  of  the  expansion  of 
—  A3  containing  02-  But  these  are  also  contained  in  —02^2- 
Hence  Ai,2  =  —  A2,  or  ^2  =  —  Ai,2. 

In  the  same  way,  by  moving  the  third  column  into  first  place 
by  two  successive  interchanges,  which  does  not  alter  the  sign  of 
the  determinant,  we  find  Ai,3  =  ^3. 

Let  Aj^k  denote  the  cof actor  of  the  element  standing  at  the 
intersection  of  the  jth  row  and  kth  column  of  A„;  we  can  bring 
this  element. to  the  intersection  of  the  first  row  and  column  by 
j  —  1  -\-k  —  1  successive  interchanges  of  rows  and  columns. 
Hence  A„  will  become  (-iy+^-2  .  A„  or  (-iy+^A„,  since  (-l)-^ 
=  1 ;  hence  by  reasoning  as  above  we  find 

11.  We  can  now  expand  An  according  to  the  elements  of  its  first 
row  in  the  form 

A„=  aiAi,i- «2Ai,2+a3Ai,3- a4Ai,4+  •  •  •   +(-l)""^Ai,„. 

To  expand  An  according  to  the  elements  of  any  other  row,  we 
can  move  this  row  into  first  place  and  then  apply  the  last  formula. 

By  this  rule  we  can  express  a  determinant  of  order  n  in  terms  of 
determinants  of  order  n  —  1.  Hence  by  repeated  application  of 
the  rule  we  can  write  out  the  complete  expansion. 

By  a  similar  process  the  determinant  can  be  expanded  ac- 
cording to  the  elements  of  any  column. 

248.  Solution  of  Systems  of  Linear  Equations.  —  We  shall  illus- 
trate the  method  of  solving  a  system  of  n  linear  equations  involving 
n  unknowns  by  considering  three  such  equations  with  three  un- 
knowns. 


248] 


DETERMINANTS 


227 


Solve  for  x,  y,  and  z  the  system  of  equations 
aiX-\-biy-}-CiZ  =  di, 
d'zX  +  62Z/  +  C2Z  =  d2, 
asx  +  hy  +  C32  =  (Zs.' 

Let  the  determinant  of  the  coefficients  be  denoted  by  A,  so  that 

tti    61    Ci 

A  =  a2  &2  C2 
as  &3  C3 

Let  the  cofactors  of  ai,  02,  as  be  Ai,  A2,  A3  respectively. 

Multiply  the  given  equations  in  order  by  Ai,  A2,  A^,  and  add  the 
results.     We  obtain 

(aiAi  +  02^2  +  03^3)  X  +(61^1  +  62A2  +  Ms)  y  + 
{ciAi  +C2A2  +  C3A3)  2  =  diAi  +  ^2^2  +  dsA-i. 

From  (7)  and  (9)  of  (247)  we  see  that  the  coefficient  of  x  is  A, 
and  of  y  and  z  zero.     Hence  we  get 


d]Ai  -\-d2A2  +  dsA: 


di 

&i 

Cl 

d2 

62 

C2 

4 

63 

Cs 

a\ 

hx 

Cl 

a2 

&2 

C2 

as 

63 

C3 

Similarly  by  multiplying  by  the  cofactors  of  61,  62,  6s  and 
adding  we  get  y,  and  by  multiplying  by  the  cofactors  of  Ci,  C2,  cs 
and  adding  we  get  z.     The  results  are  as  given  in  (241). 

In  precisely  the  same  way  we  can  solve  n  linear  equations  in 
n  unknowns. 

The  exceptional  cases  which  arise  when  A,  the  determinant  of 
the  coefficients,  is  zero,  have  been  considered  in  (242)  for  the  case 
of  two  and  three  equations.  A  similar  discussion  applies  to  the 
case  of  n  equations. 

When  the  equations  are  homogeneous  (i.e.,  c?i  =  0,  ^2  =  0, 
c^s  =  0  .  .  .),  and  A  5^  0,  the  only  solution  is  a;  =  0,  y  =  0,  2  =  0, 
.  .  .  ;  when  A  =  0,  there  exists  an  infinite  number  of  solutions. 


228 


DETERMINANTS 


[249 


249.  Exercises.     Evaluate  the  following  determinants: 


1. 


that 


a  1     3 

a  +  1     2    2 
a  +  2    3     1 

0  0  0    4 

0  0  2  10 

0  3  2     4 
6  2  0    3 

3      1      1 

1  5      0 

2  -2      1 
0      4-5 

10, 


2. 

a    h    g 

h    g   f 

9   f    c 

1   1    1 

0    0     1 

3  4-3 
1-1     4 

2,     S     4 

4  1     € 


o  a 
—  a  o 
-b  -d 
— c   —e 


h  c 

d  e 

0  J 

-J  0 


12.   Show,  without  expanding, 


1   -7 
10      5 

3  -7 


=  0. 


11. 


3. 


0  a  h 
—a  0  c 
—  b  —c    0 

6  2  8 

2  2  8 
1  6  4 

3  2  5 

0     4  4       4 

-1  -9  -1       9 

-1      7  1-1 

9    16  27     23 


Ol      0 

as     63 


0       o 
0      0 

C3      O 


Qi    bi    a    di 


13.    Show  that 
1    1    1 

X    y    z 


x2  y 


2    ^2 


(y  -  x)  (s  -  x)  (3  -  y). 


14.   Show  that 

18    36  58  50 

26    39  80  78 

17    39  55  45 

9    16  27  23 


0      4  4      4 

-1   -9  -1      9 

-1       7  1-1 

9     16  27    33 


15.  Give  two  pairs  of  values  of  x 

and  y  which  satisfy  the  equa- 
tion 

X     y    1 

3      1     1 

1  -2     1 

16.  Give  the  coordinates  of  two 
,    points  on  the  line 

y   1 

=  0. 


17.   Trace  the  graph  of 

y   1 


1    1 
-2    1 


=  0. 


18.    Give  the  coordinates  of  two 
points  on  the  line 
1  X    y    1 

I  ai  61  1    =  0. 

\         a2  ^2  1 


249] 


DETERMINANTS 


229 


19.   Give  three  sets  of  values  of  x,  y, 

z  which  satisfy  the  equation 

X      y      z     I 

3      1-21 

1-221 

•14      11 


20.   As  in  19,  for  the  equation 
y      z     \ 

Ci     a2     Qi     1 


62    bs 

C2      C3 


21.   cos  (x  +  y)  = 

23.   cos  2  X  = 
24. 


26. 


Prove  the  following  identities 
cos  X  sin  X 
sin  y  cos  y 
cos  X  sin  X  I 
I  sin  X  cos  X  I 
a         be 
sin  X  sin  y  sin  2 
cos  X  cos  2/  cos  z 
6in2x  cos^x  1 
8in2  y  cos2  ?/  1     =0. 


22.   sin  (x  —  y)  = 


sin  X  cos  X 
sin  u  cos  w 


a  sin  {y  —  z)  +  b  sin  (2  —  x)  +  c  sin  (x  —  y). 


sin2  z  cos2  z 

cos  X    sin  X  cos  x    cos  x  (s 

cosy 


27. 


+  sin  z) 
sin  y  cos  y  Cos  ?/  (sin  x  +  sin  z) 
cos  z  sin  z  cos  z  cos  2  (sin  x  +  sin  y) 
sin  X  sin  2  x  sin  3  x 
sin2  X  sin2  2  x  8in2  3  x 
sin  2  X  sin  4  X    sin  6  x 


=  0. 


2  sin  X  sin  2  X  sin  3  x  (sin  2  x 


X). 


28.  Show  that 
a  +  a'  b  +  b'  c  -\- c' 

d  e  f 

g  h  k 

29.  Show  that  the  equations 

-4X  +  2/  +  2  =  0 

are  satisfied  by 

I      1     1 
X  :  y  :  z  =  \ 


a  h  r 

a'  b'  c' 

= 

dc  f     + 

d    e   f 

g  h  k 

g    h   k 

and 


2y  + 


-4 
1 


30.   Show  that  the  equations 


are  satisfied  by 


X  :y  :  z  = 


Ix  +  my  +  nz  =0, 

[  I'x  +  m'y  +  n'z  = 

n  \    —  \l    71  \ 
'  n'\    '     \l'  n'\ 


I    7n 
I'  m' 


230  DETERMINANTS  [249 

31.  Show  that  the  equations  \3x  +  5y  +  Qz  =  0, 

[4:x  +  6y  +  7  z  =  0, 
are  eatisfied  hy  x  :  y  :  z  =  1  :  — 3:2. 

Solve  the  following  systems  of  equations: 

32.  2  x  +  3  y  -  4  3  +  7  =  0,  34.    -  r  +  s  +  t  +  u  =  4, 
7a;-4y-l=0,  r  -  s  +  t  +  u  =  S, 
9x-4z  +  l=0.  r  +  s  -  t  +  u  =  2, 

33.  20  u  +  2  r  -  7  =  0,  r  +  s  +  t-u  =  l. 

4f  +  5w;-l=0,  Z5.   2x  -  y  -3z  +  w  =  I,      . 

4w-3w  +  2  =  0.  x  +  2y  +  z-w  =  2, 

Sx-Sy-z  +  2w=-l, 
-x-y  +  2z-Zw  =  0. 

36.   k  +  l  +  m-2n  =  l, 
2k  -l  +  2m  -  4:n  =2, 
-k  +  2l  +  3m-6  7i=-2, 
k  -l  +  4:m  -8n  =-1. 


CHAPTER  XVII 

Polar  Coordinates.     Complex  Numbers.    DeMoivre's 

Theorem.     Exponential  Values  of  sin  x  and  cos  x. 

Hyperbolic  Functions 

250.  Polar  Coordinates.  —  We  have  made  repeated  use  of  the 
system  of  rectangular  coordinates,  in  which  the  position  of  any 
point  in  the  plane  is  defined  by  its  abscissa  and  ordinate.  A  second 
system  of  coordinates  defines  the  position  of  a  point  with  reference 
to  a  single  fixed  line,  called  the  initial  line,  and  a  fixed  point  on  this 
line,  called  the  origin  or  pole. 

In  the  figure,  let  OX  be  the  initial  line  and  0  the  pole.  We  shall 
consider  OX  as  the  positive  direction  of  the  initial  line.  Let  P 
be  a  point  in  the  plane  to  be 
considered.  The  position  of 
P  is  then  fixed  by  its  distance 
OF  =  r  from  0,  called  the 
radius  vector,  and  by  the 
angle  XOP  =  6,  called  the 
vectorial    angle.     Then    r,    0 

are  called  the  polar  coordinates  of  P,  and  the  point  is  indi- 
cated by  {r,d).  Similarly  Pi  is  the  point  (ri,  ^i).  The  coordi- 
nate d  is  positive  when  measured  counter-clockwise  from  OX; 
r  is  positive  when  measured  from  0  along  the  terminal  side  of  0; 
it  is  negative  when  measured  from  0  along  the  terminal  side  of  d 
produced  back  through  0.  Thus  the  points  (5,  30°)  and  (  —  5, 
210°)  coincide.     Similarly  with  (135°,  -3)  and  (-45°,  3). 


Exercise.     Plot  the  following  points: 
(45°,  1);  (45°,  -1);  (60°,  3);  (-60°,  3);  (^.  4);  ^-  "^-, 


-ITT      Z 

6"'  3 


Calculate  the  rectangular  coordinates  of  each  of  these  points,  taking  0  as 
origin  and  OX  as  the  x-axis. 

231 


232 


POLAR  COORDINATES 


[251,252 


251.  Relation  between  Polar  and  Rectangular  Coordinates. — 

Let  0  be  the  origin  and  OX  the  initial  line  of  a  system  of  polar 
coordinates  (figure).  Let  OX  and  OY 
be  the  axes  of  a  rectangular  system  of 
coordinates.     Then 

=  Va;2  +  2/2, 


^  X  =  r  cos  d, 
ly  =  rsin^; 


=  tan 


252.   Curves  in  Polar  Coordinates. 

—  When  r  and  6  are  unrestricted,  the 
point  (r,  6)  may  take  any  position  in  the  plane.  When  r  and  9  are 
connected  by  an  equation,  the  point  (r,  6)  is  in  general  restricted 
to  a  curve,  the  equation  between  r  and  6  being  called  the  polar 
equation  of  the  curve. 

Example  1.     Trace  the  curve  whose  polar  equation  is  r  =  sin  d. 
Assume  a  series  of  values  for  6,  calculate  the  corresponding  values  of  r  and 
plot  the  points  whose  coordinates  are  corresponding  values  of  r  and  6. 


0°,  30°,  60°,  90°,  120°,  150°,  180°,  210°,     240°,      270°, 
0,    .5,  .87,   1.0,    .87,       .5,       0,    -.5,    -.87,     -1.0, 


The  graph  is  shown  in  the  figure. 
For  values  oi  0  >  360°,  and  for 
negative  angles,  no  new  points  are 
obtained.  The  curve  is  a  circle, 
with  radius  =  ^. 

Example  2.  Trace  the  curve 
r  =-2  0. 

Here  0  is  understood  to  be  in 
radians. 


300° 

-.87, 


330°, 
-.5, 


360°. 
0. 


r  =  0,-,  7r,-j,2^,.. 

For  negative  values  of  0  we 
get  corresponding  negative 
values  of  r.  The  curve  is 
the  double  spiral  in  the  fig- 
ure, the  branches  shown  by  the  full  line  and  the  dotted  line  being  obtained 
from  the  positive  and  the  negative  values  of  6  respectively. 


263 


COMPLEX  NUMBERS 


233 


Exercises.    Trace  the  following  curves: 

1.  r  =  2  sin  d.  6.   r  =  sin- 1  e. 

2.  r  =  cose .  6.   r  =  tan- 1  e. 

3.  r  =  tan  d.  7.   rO  =  1. 

4.  r  =  sec  0.  8.   r  =  2*. 


10.  r  =  logio  6. 

11.  r  =  4. 

12.  9=^ 


71/ 


253.  Complex  Numbers.  —  Let  a  and  h  denote  any  two  real 
numbers  and  i  =  V—  1.  Then  the  quantity  a  +  ih  is  called  a 
complex  yiumher.  It  may  be  considered  as  made  up  of  a  real 
units  and  h  imaginary  units,  a  X  1  +  6  X  t. 

Real  numbers  can  be  represented  by  points  on  a  straight  line. 
To  represent  complex  numbers  ^y 
geometrically,  we  require  a  plane. 
Let  OX  and  OF  be  a  system  of 
rectangular  axes,  and  P  a  point 
in  their  plane  having  coordinates 
(a,  6)  (figure).  Then  P  is  called 
the  representative  point  of  the 
complex  number  a -\- ih. 

When  5  =  0,  P  lies  on  the  a;-axis,  and  the  complex   number 
reduces  to  a  real  number.     Thus  all  points  on  the  x-axis  corre- 
spond to  real  numbers,  and  this 
line    is    called   the   axis   of   real 
numbers. 

Let  P  (figure)  be  a  point  {x,  y) 
in  the  plane,  and  let  z  be  the  com- 
plex number  represented  by  P. 
Then 

z  =  oc-^iy. 

Now  take  OX  as  the  initial  line  and  0  as  the  pole  of  a  system  of 
polar  coordinates.    Let  the  polar  coordinates  of  P  be  (r,  d).     Then 


Hence 


X  =  r  cos  6;  y  =  r  sin 


z  =  X  -\-  iy  =  r  (cos  6  -^  i  sin  6) . 


Here  r  is  called  the  modulus  and  d  the  angle  of  the  complex 
number  z. 


234 


COMPLEX  NUMBERS 


[  254,  255 


When  r  is  fixed,  and  6  is  changed  by  integral  multiples  of  2  tt, 
we  obtain  a  set  of  complex  numbers  of  the  form, 

z  =  r  [cos  {d+2mv)  -\-i  sin  {d-\-2  mr)\) 

n  =  Q,  ±1,  ±2,  .  .  .     . 

All  these  numbers  have  the  same  representative  point. 

254.  Addition  of  Complex  Numbers.  —  The  sum  of  two  com- 
plex numbers, 

z  =  x-\-iy     and     z'  =  x'  +  iy', 

we  define  by  the  equation 

z -\- z'  =  {x -\- x')  +  i  (y  +  y'). 

We  proceed  to  consider  this 
sum  geometrically.  Let  P,  P' 
(figure)  be  the  representative 
points  of  z,  z'  respectively.  On 
OP  and  OP'  as  adjacent  sides  con- 
struct the  parallelogram  OPQP'. 
Then  Q  is  the  representative  point 
of  z  -\-  z' .  For  the  coordinates  of 
Q  are  {x -\- x' ,  y  +  y'). 
The  difference  of  the  two  complex  numbers  z  and  z'  we  may 
define  by  the  equation 

z-  z'  =  {x-  x')  +i{y  -  ij'). 

Exercise.    Give  a  geometric  construction  for  the  representative  point  of 
z  —  z' . 

255.  Multiplication  of   Complex  Numbers.  —  The  product  of 
the  two  complex  numbers, 

z  =  r  (cos  d  -\-i  sin  6)     and     z'  =  r'  (cos  6'  +  i  sin  6'), 

we  define  by  the  equation 

zz'  =  rr'  (cos  6  -\- i  sin  6)  (cos  6'  -{- i  sin  6'). 


/ 

256,257]  DE  MOIVRE'S  THEOREM  235 

Multiplying  out  the  product  of  the  two  binomials  wc  find 

zz'  =  rr'  [cos  d  cos  6'  —  sin  d  sin  0'  +  i  (sin  d  cos  6'  -\-  cos  6  sin  d')\ 
=  rr'  [cos  {e  +  0')  +  i  sin  (9  +  ^')]- 

Therefore  the  modulus  of  the  product  zz'  equals  the  product  of  the 
moduli  of  z  and  z',  and  the  angle  of  zz'  equals  the  sum  of  the  angles 
of  z  and  z'. 

By  repeating  this  process  we  find 

zz'z"  .  .  .    =  rr'r"  •  •  •  [cos  {9  -\-  6'  -\-  6"  +   •  •  -  ) 
-j-ism{d-\-d'  -{-d"  +   •  •  •  )] 

for  any  finite  number  of  factors  z,  z',  z",  .... 
When  the  factors  are  all  equal  this  reduces  to 

z"  =  r"  (cos  nQ  +  i  sin  uQ), 

n  being  a  positive  integer. 

Exercise.    Show  that  the  above  definition  of  the  product  zz'  is  the  same  as 

zz'  =  xx'  -  yy'  +  i  (xy'  +  x'y), 
where  z  =  x  -\-iy     and     2'  =  x'  +  iy'- 

256.  De  Moivre's  Theorem.  —  When  r  =  \,  then  z  =  cos  6  + 
i  sin  6.     Hence  by  the  above  result  we  have 

(cos  6  +  i  sin  6)"  =  cos  nS  +  *  sin  «6. 

This  equation  contains  what  is  kno\\Ti  as  De  Moivre's  Theorem. 

257.  Definition  of  z^\  —  Let  p  be  any  real  number,  positive  or 
negative,  rational  or  irrational.  Then  by  analogy  with  the  result 
for  2''  when  n  is  a  positive  integer,  we  define  zp  by  the  equation 

zP  =  rP  (cos  i>6  +  i  sin  p^), 

where  z  =  r  (cos  6  -\-  i  sin  6). 

Then,  if  q  also  be  real,  we  have 

zi  =  7-9  (cos  qd  -\-  i  sin  qO), 
and 

^v^q  ^  ;.p+g[cos  {p-\-q)d  +  i  sin  (p  +  q)d]=  z^^^. 


236 


COMPLEX  NUMBERS 


[258 


Hence  the  rules  for  exponents  will  be  the  same  when  the  base  is  a 
complex  number  as  when  the  base  is  real. 
Examples. 

1.   Find    the    modulus   and   angle    of   2  =  3 
-4i. 
Here  3  =  r  cos  0;    —  4  =  r  sin  0. 

r  =  V32  +  42  =  5;  tan  »  =  ^ . 


i-t)- 


The  angle  lies  in  the  fourth  quadrant. 

2.   Express  2  (cos  150°  -  i  sin  150°)  in  the 
form  X  +iy. 


2  (cos  150° 


ism  150°)  =2(  -^  V3  -^ 


3.   Find  the  value  of  (1  +  i)^  (2-3  i). 

{l+iy  =  l+2i  +  i^  =  2i. 
(1  +  i)M2  -  3  i)  =  2  i  (2  -  3  i)  =  4  i  -  6  i2 : 


V3-i. 


+  4  i. 


Exercises. 

1.  Find  the  modulus  and  angle  of 

1-i;    4  +  3i;    -5  + Hi;    2i;    2;     (l+i)(l-t); 
3V3+3i;     (3V3-3i)^     {l +i^3)(\/S +i). 
Give  figure  for  each  case. 

2.  Find  the  value  of  : 

(l+i)3;     (l-i)i;     (l+i)2(l+2i)2;     (3-3i)2  (VS  +  O^      (l-i^/s)\ 

258.   Theorem.     If  P  and  Q  are  any  real  quantities  and  if 
pj^iQ=  0,  then  P=0  andQ  =  0. 

Proof.    By  hypothesis,  P  +  iQ  =  0    or     P  =  -  iQ. 
Squaring,  p2  =  _  Q2, 

Now  P2  and  Q^  must  be  positive,  hence  the  last  equation  states 
that  a  positive  quantity  equals  a  negative  quantity.  This  is 
impossible  unless  both  quantities  are  zero. 

P  =  0    and    Q  =  0. 
This  theorem  is  used  to  replace  a  given  equation  of  the  form 

P  +  tQ  =  0 
by  the  equivalent  equations 

p  =  0;    Q  =  0. 


259] 


ROOTS  OF  UNITY 


237 


As  a  corollary  we  have,  if 

P-\-iQ  =  P'  +  iQ', 
then  P  =  P'    and    Q  =  Q'. 

For  the  given  equation  is  equivalent  to  (P  —  P')  -\-i{Q  —  Q')  =  0. 
259.   The  nth  Roots  of  Umt^ — To  solve  the  equation 
X"  -  1  =  0,     or    X"  =  1, 
replace  1  by  its  value  cos  2  kw  +  i  sin  2  kir,  k  being  an  integer. 

We  obtain 

re"  =  cos  2kT  -\-  i  sin  2  kir. 

Jaking  the  nth  roots  of  both  members  we  have,  by  putting  P  =  - 

in  (257), 

2k7, 


,    .   .    2kir 

X  =  cos h  1  sm 

?i  n 


Here  fc  may  be  any 
integer;  letting  k 
=  0,  1,  2,  .  .  . 
n  —  1,  we  obtain  n 
distinct  values  of 
X,  that  is,  n  dis- 
tinct nth  roots  of 
1.  For  other  values 
of  k  we  obtain  the 
same  roots  over 
again. 

Geometric  Rep- 
resentation of  the 
nth  Roots  of  Unity. 
—  The  nth  roots  of 
1  are, 


fc  =  0;     Xi  =  cos  0  +  i  sin  0=1, 

7         1  27r    ,     .    .     27r 

fc  =  1 ;     Xo  =  cos h  *  sm  — > 

n  n 

TO                     47r   ,    .   .    47r 
A;  =  2:     Xs  =  cos h  ^  sm  —  > 


2  (n  -  1)  TT   ,    .   .    2  (n  -  1)  7r 
fc  =  n  —  1 ;     a-^  =  cos  ^ 1-  i  sm 


238 


EXPANSION  OF  SIN  nd  AND  COS  ni 


[260 


The  representative  points  of  Xi,  X2,  xs,  .  .  .  x„  are  ob- 
tained as  n  equally  spaced  points  on  a  circle  of  radius  1,  the 
coordinates  of  the  first  point  being  (1,0)  (figure). 

To  obtain  the  nth  roots  of  any  number  a,  we  need  only  multiply 
one  of  its  arithmetic  nth  root  by  the  nth  roots  of  unity. 

Example.     Find  the  cube  roots 
of  unity. 
These  are  given  by 

x  =  cos— 5 — hisin-i^;     fc  =  0,1,2. 


/c  =  0; 

xi  =  cos  0°  +  *  sin  0°  =  1. 

ft. 

h  =  \; 

X2  =  cos  120°  +  i  sin  120° 

-i+iv3. 

k  =2.; 

X3  =  cos  240°  +  i  sin  240° 

To  find  the  cube  roots  of  8,  we  have  -^/S  =  2  'x/l  =  2;  - 1  +  i  V3;  - 1  -  i  Vs. 
(We  here  use  v'S  to  denote  any  cube  root  of  8,  not  merely  the  principal  root.) 

Exercises. 

1.  Solve  the  equations  x^  -  1  =0  and  a;^  -  8  =  0  algebraically  and  com- 
pare with  above  results. 

Solve  the  following  equations  by  the  trigonometric  method  and  give  a  figure 
for  each  case: 

2.  x4  =  1 ;  4.   x5  =  1 ;  6.   x^  =  1 ; 

3.  x4  =  81;  5.   x5  =  32;  7.   x^  =27. 

260.   To  express  sin  n6  and  cos  nQ  in  terms  of  powers  of  sin  6 
and  cos  6,  w  being  a  positive  integer. 
We  have 

(cos  6  -\-  i  sin  6Y  =  cos  nd  -f  i  sin  nd. 

Expand  the  left  member  by  the  binomial  theorem,  reduce  all 
powers  of  I  to  ±1  or  ±i,  and  group  the  real  terms  and  those 
involving  i.     The  above  equation  then  becomes 


COS  nd  +  i  sin  nd 
-\-  iln  cos"~^ 


|2 
n  {n  —  l)(n 


2) 


^ 


2G1]        EXPONENTIAL  VALUES  OF  SIN  X  AND  COS  X       239 

This  equation  has  the  form 

P  +  iQ  =  P'  +  iQ'. 

Hence  by  the  corollary  in  (258)  we  have 
cos  nQ  =  cos"  d  -  ^     |~      cos"--  6  sin^  d -\-  -  -  -  . 
sin  nd  =  n  cos"-  ^  g  sin  0  -  "^  ^"^  ~  j^^'^  ~  ^^  cos"-^  6  sin^  g  +  •  •  -. 

sin  4  9  =  4  cos3  0  sin  i9  -  4  cos  0  sin^  0. 

cos  5  19  =  cos5  0-10  cos3  d  sin2  e  +  5  cos  ff  sin^  e. 

Exercises.     Expand  in  powers  of  sin  9  and  cos  9: 

1.  sin  39;  3.    cos  40;  6.   sin  60; 

2.  cos30;  4.   sin  5  0;  6.    cos70. 

261.   Exponential  Values  of  sin  £c  and  cos  a?.  —  We  have  the 
expansions,  (219), 

e^  =  l  +  a:  +  |  +  |+  •  •  •  ; 

sinx  =  a:-^  +  ^-   •  •  •  , 


■       cosx=l-|2  +  |4-   •  •  •   • 

In  the  first  series  replace  x  by  ix  and  define  the  result  to  be  e*^: 
noting  that 

i2  =  _  1    ^-3  =  -  i,  i^  =  1,  •  -  -  , 


we  obtain 

eix  =  1  + 

a:2 

4^r^- 

-    •  • 

4- 

12+ L* 

-  •  ■ 

>,•(.- 

S* 

X5 

|5 

Hence 

> 

e'-  = 

coscc 

+  i  sin  X. 

Replacing  x 

by  -x; 

=  cos  a?  —  i  sin  x. 


240  HYPERBOLIC  FUNCTIONS  [262 

From  these  equations  we  find 


3  i 


These  formulas  are  useful  in  many  appHcations  of  the  trigono- 
metric functions. 

Exercises.     Using  the  exponential  values  of  sin  x  and  cos  x,  show  that: 

1.  sin2  X  +  cos2  X  =  I.  3.    cos  2  a;  =  cos^  x  —  sin2  x. 

2.  sin  2  X  =  2  sin  x  cos  x.  4.    cos*  x  —  sin'*  x  =  cos^  x  —  sin^  x. 

262.  The  Hyperbolic  Functions.  —  In  the  expansions  for  sin  x 
and  cos  x  given  at  the  beginning  of  (261)  replace  x  by  ix  and  define 
the  results  to  be  sin  ix  and  cos  ix  respectively.     We  obtain 


smix 


x^ 


cos  ix  =  1  +  ^  +  T^  +.    •  .  •  . 

These  equations  we  consider  as  defining  the  sine  and  cosine  of 
the  imaginary  quantity  ix. 

Multiply  the  first  equation  by  i  and  subtract  the  result  from  the 
second.     We  obtain 

cos  ix  —  i  sin  ix  =  e*. 

Change  a;  to  —a:; 

cos  ix  -\-  i  sin  ix  =  e~*. 

(Note  that  sin  ix  =  -  sin  (-  ix)  by  the  definition  of  sin  ix.) 
Combining  the  last  two  equations  by  addition  and  subtraction, 
we  find 

cos  IX  = ^ ;  sm  IX  =  I 


2        '  2 

We  now  define 

Hyperbolic  cosine  of  x  (=  cosh x)  =  cos  ix; 
Hyperbolic  sine  of  x  ( =  sinh  x)  =  -  sin  ix.    ^ 


Then 


cosh  a?  = ,    sinh  a?  = ^ — 


262]  HYPERBOLIC  FUNCTIONS  241 

These  functions  are  related  to  the  hyperbola  somewhat  as  the 
circular  functions  to  the  circle. 

The  remaining  hyperbolic  functions  are  defined  by  the  equa- 
tions 

sinh  a? .       ^,  1       . 

coth  a?  =  :: — ; —  ; 


tanh 

coshic 

sech 

1 

cosh  a; 

Exercises. 

Show  that: 

1. 

sinhO 

=  0; 

cosh  0  =  1. 

2. 

sinhiri 

=  0; 

cosh  irl  =  —  ] 

3. 

sinh- 

=  i; 

cosh|'=0. 

csch  £c  = 


sinh 


6.  cosh  ( —  x)  =  cosh  x. 
^-             6.  cosh2  X  -  sinh2  x  =  1. 

7.  sech2  X  =  1  —  tanh2  x. 

4.   sinh  ( —  x)  =  —  sinh  x.  8.    —  csch2  x  =  1  —  coth2  i. 

Draw  the  graphs  of  the  equations  (see  tables) : 

9.    y  =  e^.  11.    y  =  cosh  x. 

10.  y  =  e-^.  12.   2/=  sinh X. 


CHAPTER  XVIII 

Permutations.    Combinations.    Chance 

263.  Permutations.  —  A  -permutation  is  a  definite  order  or 
arrangement  of  a  group  of  objects,  or  of  part  of  the  group*. 

Let  there  be  a  group  of  n  distinct  objects.  The  number  of 
possible  arrangements,  taking  r  of  these  objects  at  a  time  is  called 
the  number  of  permutations  of  n  things  r  at  a  time,  and  is  denoted 
by  „P,. 

Theorem  1.     The  number  of  permutations  of  n  things  r  ata  time  is 

«Pr  ^  n{n-l)   .   .   .    (»i  -  i>  4-  1). 

Proof.     Evidently  „Pi  =  n. 

Now  with  each  of  the  n  objects  we  may  pair  any  one  of  the  remain- 
ing n  —  1  objects. 

Hence  „P2  =  n(n  —  1). 

With  each  one  of  these  n{n  —  1)  permutations  containing  2  objects 
we  may  associate  one  of  the  remaining  n  —  2  objects. 

Hence  nPz  =  n{n  —  I)  {n  —  2). 

Proceeding  in  this  way  we  obtain  the  formula  stated. 
When  r  =  n  we  have 

nPn  =  I  n. 
Exercises. 

1.  How  many  numbers  of  four  figures  each  can  be  formed  from  the  digits 
1,  2,  3,  4  ? 

2.  How  many  3-figure  numbers  can  be  formed  from  the  digits  1,  2,  3,  4,  5? 
y;  3.  How  many  numbers  greater  than  1000  can  be  formed  from  the  digits 
i,  3,  5,  7,  9? 

4.    How  many  changes  can  be  rung  with  8  bells,  4  at  a  time? 

264.  Combinations.  —  A  combination  is  a  group  of  objects, 
without  reference  to  their  arrangement. 

242 


265]  PERMUTATIONS  AND  COMBINATIONS  243 

The  number  of  different  groups  or  combinations  of  n  objects, 
each  group  containing  r  objects,  is  called  the  number  of  combina- 
tions of  /)  things  r  at  a  time,  and  is  denoted  by  „Cr. 

Theorem  2.  The  number  of  combinations  of  ?i  things  r  at  a 
time  is 

_  „F,.  _  n(n  -1)   ■   •   •   {n  -  r  +  1) 

Proof.  Suppose  all  the  combinations  of  the  n  things  r  at  a  time 
to  be  written  down.  Each  group  so  written  will  yield,  by  per- 
muting its  objects  in  all  possible  ways,  |_r  permutations.  Hence 
there  are  \r  times  as  many  permutations  as  combinations,  or 

\r^nCr  =  nPr  =  ^  (n  -   1)     .    .    .     (^i  -  r  +  1). 

Hence  the  theorem. 

Exercises. 

1.  How  many  triangles  can  be  formed  from  6  points,  no  three  points  being 
collinear? 

2.  How  many  tetrahedrons  can  be  formed  from  12  points,  no  four  points 
being  coplanar? 

3.  How  many  committees  of  3  persons  each  can  be  formed  from  a  club 
of  10  persons? 

4.  Show  that  nCr   =  rvCn-r- 

(This  is  a  convenient  formula  when  r  is  nearly  as  large  as  n.  It  is  then 
shorter  to  calculate  nCn-r-) 

6.    Show  that  „Co  +  nCl  +  «C2  +   •   •  •   +  nCn  =  2". 
(Expand  (1  +  x)"  and  put  x  =  1;  nCo  is  defined  to  be  1.) 

6.  How  many  committees,  consisting  of  from  1  to  9  members,  can  be 
formed  from  a  club  of  10  persons? 

7.  Find  the  value  of  20C18. 

265.  Theorem  3.  —  The  number  of  permutations  of  n  things  n  at 
a  time,  when  p  things  are  alike,  is 

\n 

\P 

Proof.  Let  P  be  the  number  of  permutations  sought,  and  sup- 
pose them  written  down.  If  now  the  p  things  in  question  were 
unlike,  by  permutating  them  among  themselves  each  of  the  P 
permutations  would  yield  \p  permutations;  the  total  number  of 
permutations  so  formed  would  be  |^  P  and  must  equal  n^n  or  [n. 
Hence  the  theorem. 


244  CHANCE  [266,267 

Similarly,  the  number  of  permutations  of  n  things  n  at  a  time, 
when  p  things  are  all  of  one  kind,  and  g  of  a  second  kind,  will  be 

\n 

\p\q 
and  so  on. 

266.  Exercises. 

1.  How  many  permutations  of  seven  letters  each  can  be  formed  from  the 
letters  of  the  word  "arrange"? 

2.  How  many  permutations  of  11  letters  each  can  be  formed  from  the 
letters  of  the  word  "Mississippi"? 

3.  How  many  words,  each  containing  a  vowel  and  two  consonants,  can 
be  formed  from  4  vowels  and  6  consonants? 

4.  How  many  even  numbers  of  four  figures  each  can  be  formed  from  the 
digits  1,  2,  3,  4,  5,  6? 

5.  How  many  elevens  can  be  chosen  from  20  players  if  only  6  of  the  20 
are  qualified  to  play  behind  the  line? 

6.  As  in  5,  if  in  addition,  only  2  men  are  qualified  for  center. 

7.  How  many  sums  can  be  formed  with  one  coin  of  each  denomination, 
from  a  cent  to  a  dollar? 

8.  As  in  7,  except  that  there  are  two  coins  of  each  denomination. 

9.  If  two  coins  are  tossed,  in  how  many  ways  may  they  fall? 

10.  As  in  9,  for  10  coins. 

11.  If  two  dice  are  thrown,  in  how  many  ways  may  they  turn  up? 

12.  As  in  11,  for  3  dice. 

267.  Probability  or  Chance.  —  If  a  bag  contains  4  white  and 
3  black  balls,  and  a  ball  is  drawn  at  random,  what  is  the  chance 
that  it  be  white  ? 

In  order  to  solve  this  problem  we  first  define  chance  or  proba- 
bility. 

Definition.  The  measure  of  the  probability  of  the  occurrence  of 
an  event  is  taken  to  be  the  quotient, 

number  of  favorable  ways 
total  number  of  possible  ways 

In  the  problem  above,  since  there  are  7  balls  altogether,  there  are 
7  possible  ways  of  drawing  one  ball ;  of  these  4  are  favorable,  since 
there  are  4  white  balls.     Hence  the  chance  that  a  white  ball  be 

drawn  is  =  • 

^  3 


Similarly  the  chance  for  a  black  ball  is  =• 


2G7]  .     CHANCE  245 

If  an  event  can  happen  in  a  ways,  and  fail  in  b  ways,  then,  by  the 

definition,  the  chance  that  it  will  happen  is  — r-^,  and  that  it 

a  +  6 

will  fail  is 


a  +  6 
Since  the  event  must  either  happen  or  fail,  the  probability 

for  which  is  — -^  -j :— 7-  =  1 ,  we  have  1  as  the  mathematical 

a-\-b      a  +  6 

symbol  for  certainty. 

If  p  is  the  probability  that  an  event  will  happen,  1  —  p  is  the 

probability  that  it  will  fail. 

Example  1.     From  a  bag  containing  4  white  and  3  black  balls,  2  balls  are 

drawn  at  random. 

(a)  What  is  the  chance  that  both  be  white? 

Number  of  favorable  ways:     4C2  =  6. 

Number  of  possible  ways:       7(^2  =  21. 

6       2 
Hence  the  required  chance  is:       p  =  ^  =  ^• 

(6)  What  is  the  chance  that  at  least  one  be  white? 
Favorable  cases:  both  white,  4(^2  =  6; 

one  white,  other  black,  3  X  4  =  12. 
.'.    Total  number  of  favorable  cases  is  18. 
Number  of  possible  cases,  as  before,  21. 
18      6 
Hence  ^  ""  21  ""  7 ' 

A  shorter  method  is  as  follows:  The  probability  that  both  balls  be  black 

is  ?^  =  —  =  - .     Hence  the  chance  that  at  least  one  be  white  is  1  -  =  =  ^^  • 
7C2      21       7  <       ' 

Exam-pie  2.     From  12  tickets,  numbered  1,  2,  .  .  .  12,  four  are  drawn  at 

random. 

(o)  What  is  the  probability  that  they  bear  even  numbers? 

Since  6  tickets  bear  even  numbers,  the  number  of  favorable  cases  is  6C4. 

The  total  number  of  ways  of  drawing  4  tickets  from  12  is  12C4.     Hence 

=  ^=     6-5-4.3     ^2 
^       12C4       12-11 -10 -9      33' 

(6)  What  is  the  chance  that  two  bear  even,  the  other  two  odd  numbers? 

We  can  select  two  tickets  bearing  even  numbers  in  gG  ways;  also  two  bear- 
ing odd  numbers  in  6^2  ways.     Combining  any  one  of  the  first  with  any  one 
of  the  second  gives  6C2  X  &C2  favorable  ways.     Hence 
6C2  X  6C2       5 


V 


12C4  11 


246  CHANCE  [268,269 

268.  Exercises. 

1.  If  5  coins  are  tossed,  what  is  the  chance  of  three  heads? 

2.  If  5  coins  are  tossed,  what  is  the  chance  of  at  least  two  heads? 

3.  If  3  balls  are  drawn  from  a  bag  containing  5  white  and  4  black  balls, 
what  is  the  chance  that  all  three  are  white? 

4.  In  exercise  3,  what  is  the  chance  of  drawing  2  white  balls  and  one 
black  ball? 

5.  In  exercise  3,  what  is  the  chance  of  drawing  at  least  one  white  ball? 

6.  What  is  the  chance  of  two  sixes  in  a  single  throw  of  two  dice? 

7.  What  is  the  chance  of  throwing  three  sixes  in  a  single  throw  with  three 
dice? 

8.  Three  dice  are  thrown.     What  is  the  chance  that  the  sum  of  the 
numbers  turned  up  is  11? 

9.  As  in  8,  except  that  the  sum  is  to  be  7. 

10.  Six  cards  are  drawn  from  a  pack  of  52.  What  is  the  chance  of  three 
aces? 

11.  Six  cards  are  drawn  from  a  pack  of  52.  What  is  the  chance  that  all 
are  of  the  same  suite? 

269.  Compound  Probabilities. 

Definition.  Two  events  are  said  to  be  independent  when  the 
occurrence  of  one  does  not  affect  that  of  the  other. 

Theorem  4.  The  chance  that  both  of  two  independent  events  shall 
happen  is  the  product  of  their  separate  probabilities. 

Proof.  Suppose  the  first  event  happens  in  a  ways  and  fails  in 
h  ways,  out  of  a  a  +  6  possible  ways,  and  that  the  second  happens 
in  a'  ways  and  fails  in  b'  ways,  out  of  a  total  of  a'  +  6'  ways. 

Combining  each  of  the  a  favorable  ways  of  the  first  event  with 
each  of  the  a'  favorable  ways  of  the  second,  we  have  aa'  favorable 
cases.  The  total  number  of  possible  cases  is  (a  +  b)  (a'  +  6')- 
Hence 

_  ^^'  _      ^      V      ^' 

^  ~  (a  +  6)  (a'  +  b')  ~  M^      a^T&^ 

which  is  the  product  of  the  separate  probabilities  of  the  two  events. 

As  an  immediate  extension,  we  have  the 

Theorem  5.  //  the  probabilities  of  several  independent  events  be 
Pi,  P2,  '  '  •  Vnj  ihe  probability  that  all  will  happen  is 

P   =Pl   Xp-iX      '     '     '      XPn. 

Example.  From  a  bag  containing  4  white  and  3  black  balls,  2  balls  are 
drawn  in  succession.     What  is  the  chance  that  both  are  white? 


269]  CHANCE  247 

On  the  first  drawing  the  chance  for  a  white  ball  is  ^ ;  on  the  second,  ^ .  The 
probabihty  of  both  events  is  therefore 

7^6      7 

Definition.  Two  events  are  said  to  be  dependent  when  the 
occurrence  of  one  of  them  affects  that  of  the  other. 

Theorem  6.  Of  n  dependent  events,  let  the  chance  that  the  first 
will  happen  he  p\,  the  chance  that  the  second  then  follows  he  p2,  that 
the  third  then  follows  he  ps,  and  so  on.  The  chance  that  all  these 
events  shall  happen  is  then 

P  =  PiXp2Xp3  X   •   ■  '  Pn 

This  is  an  immediate  consequence  of  the  preceding  theorem. 

Theorem  7.  If  p  be  the  chance  that  an  event  will  happen  in  one 
trial,  the  chance  that  it  will  happen  just  r  times  in  n  trials  is 

Proof.  The  chance  that  r  trials  out  of  n  shall  succeed  is  p*", 
and  that  the  other  n  —  r  trials  shall  fail  is  (1  —  pY'"".  Hence  the 
probability  of  success  in  r  particular  trials  and  of  failure  in  the 
n  —  r  other  trials  is  p"  {\  —  pY~\  But  of  the  n  trials,  any  r 
may  be  the  successful  ones,  which  gives  „Cr  possibilities,  each 
having  a  probability  p''  (1  —  ?))""'".     Hence  the  result  stated. 

Examples. 

1.  In  a  class  of  3  students,  A  solves  on  the  average  9  problems  out  of  10, 
B  8  out  of  10,  C  7  out  of  10.  What  is  the  chance  that  a  problem,  presented 
to  the  class,  will  be  solved? 

The  problem  will  be  solved  unless  all  three  students  fail,  the  probability 
for  which  is 

10      10      10      500 

Hence  the  chance  that  the  problem  will  be  solved  is 

_  A^MZ 

500      500' 

2.  Two  bags  each  contain  5  black  balls,  and  a  third  bag  contains  5  black 
and  5  white  balls.  What  is  the  chance  of  drawing  a  white  ball  from  one  of 
the  bags  selected  at  random? 


248  CHANCE  [270 

The  chance  that  the  bag  containing  white  balls  be  chosen  is  5 .    The  chance 

1  "^ 

that  a  white  ball  be  now  drawn  from  this  bag  is  ^ .     Hence  the  probabihty 

that  both  events  happen  and  that  a  white  ball  be  drawn  is 

3      2      6 
3.   A  coin  is  tossed  10  times.     What  is  the  chance  for  just  3  heads? 
The  probability  of  a  head  in  one  trial  is  ^  •     Hence 


nCrV^  (1  -p)"-'-   =  loCsQJ^l  -0  = 

270.  Exercises. 


128' 


1.  Three  hats  each  contain  5  tickets,  those  in  two  of  the  hats  being  num- 
bered 1,2,  ...  5,  and  those  in  the  third  hat  being  blank.  What  is  the  chance 
of  drawing  a  ticket  bearing  an  even  number  from  one  of  the  hats  selected  at 
random? 

2.  If  in  exercise  1  two  tickets  be  drawn  from  a  hat  chosen  at  random, 
what  is  the  chance  that  both  bear  even  numbers? 

3.  If  each  of  two  persons  draw  a  ticket  from  one  of  the  hats  in  exercise  1, 
the  first  ticket  being  replaced  before  the  second  is  drawn,  what  is  the  chance 
that  both  persons  draw  the  same  number?  What  is  the  chance  that  both 
draw  blanks? 

4.  If  a  coin  be  tossed  10  times,  what  is  the  chance  for  at  least  7  heads? 

5.  How  many  different  sets  of  throws  can  be  made  with  a  coin,  each  set 
consisting  of  5  successive  throws? 

6.  The  chance  that  a  person  aged  25  years  will  live  to  be  75  is  2;| .  What  is 
the  chance  that,  of  three  couples  married  at  the  age  of  25,  at  least  one  shall 
live  to  celebrate  their  golden  wedding? 

7.  A  bag  contains  10  white,  6  black,  and  4  red  balls.  Find  the  chance  that, 
of  three  balls  drawn,  there  shall  be  one  of  each  color. 

8.  A  gunner  hits  the  target  on  an  average  7  times  out  of  10.  What  is 
the  chance  that  5  consecutive  shots  shall  hit  the  target? 

9.  Two  dice  are  thrown.  Find  the  chance  that  the  sum  of  the  numbers 
turned  up  shall  be  even. 


CHAPTER  XIX 

Theory  of  Equations 

271.  We  shall  refer  to  the  equation 

(1)       i^o-^" +i>i^"-'+i>2x"-^+   •  •   •   +i>„-ia7+i>„  =0 

as  the  standard  form  of  the  equation  of  « th  degree ;  pox"  is  called 
the  leading  term  and  p„  the  constant  (or  absolute)  term. 

The  coefficient  of  the  leading  term  may  be  made  equal  to  unity 
by  dividing  the  whole  equation  by  this  coefficient. 

When  all  the  terms  written  in  equation  (1)  are  present,  the 
equation  is  said  to  be  complete;  when  one  or  more  terms  are 
absent,  the  equation  is  said  to  be  incomplete.  An  incomplete 
equation  may  be  made  complete  by  supplying  the  missing  terms 
with  zero  coefficients. 

We  shall  represent  the  polynomial  forming  the  left  member  of 
equation  (1)  hyf{x);  f  (a)  shall  denote  the  value  of  this  poly- 
nomial when  X  =  a,f  (b)  the  value  when  x  =  h,  and  so  on. 

A  root  of  an  equation  is  a  value  of  x  which  satisfies  the  equa- 
tion; hence  a  is  a  root  of  the  equation  /  (x)  =  0  if  /  (a)  =  0. 

In  the  present  chapter  we  shall  consider  methods  of  finding  the 
roots  of  the  equation  /  (x)  =  0. 

272.  Factor  Theorem.  —  If  a  is  a  root  of  the  equation  f  (x)  =  0, 
thenfix)  is  divisible  by  (x  -  a),  and  conversely. 

Proof.  Divide  /  (x)  by  {x  -  a) ;  let  Q  be  the  quotient,  R  the 
remainder.     Then 

f{x)  =  {x-a)Q  +  R. 

Putting  X  =  a,  we  obtain  R  =  0,  since  /  (a)  =  0  by  hypothesis. 
Hence  /  (x)  is  divisible  by  {x  -  a)  without  a  remainder. 
Conversely,  assume 

f(x)  =  (x-a)Q. 

Fxit  x  =  a  and  we  have  /  («)  =  0;  hence  a  is  a  root  of  /  (x)  =  0. 

[See  also  (11),  (f).] 

249 


250  THEORY  OF  EQUATIONS  [273-275 

273.  Depressed  Equation.  —  When  a  is  a  root  of  the  equation 
/  (x)  =  0,  we  may  write 

f(x)  =  {x-a)Q. 
Any  other  value  of  x  which  reduces  /  (x)  to  zero  must  also  reduce 
Q  to  zero,  and  is  therefore  a  root  of  the  equation  Q  =  0. 
.  But  if/  (x)  is  of  degree  n,  Q  will  be  of  degree  n  —  1.  Hence  when 
one  root  of  an  equation  is  known,  the  other  roots  may  be  found 
by  solving  an  equation  of  degree  one  less  than  that  of  the  given 
equation,  and  whose  left  member  is  found  by  dividing  the  left 
member  of  the  given  equation  by  {x  —  the  root). 

The  new  equation  is  called  the  depressed  equation. 

Exercises.  Show  that  each  of  the  following  equations  has  the  root  indi- 
cated, and  find  the  other  roots: 

1.  x3  -  9x2  +  26a;  -  24  =  0;    x  =  2. 

2.  x4  +  3a:2_8x -24  =  0;     x  =  -  3. 

3.  3x3-14x2  +  17x-6  =  0;    x  =  f . 

274.  Number  of  Roots.  —  We  assume  that  every  equation  of 
the  form  (1),  (271),  has  at  least  one  root,  that  is,  there  exists  at 
least  one  value  of  x,  real  or  imaginary,  which  satisfies  the  equation. 
It  can  then  be  shown  that  an  equation  of  the  nth  degree  has  just 
n  roots. 

For,  let  ai  be  a  root.  Form  the  depressed  equation  by  removing 
from  /(x)  the  factor  x  -  ai,  and  let  the  new  equation,  of  degree 
n  —  1,  he  fiix)=  0.  By  the  above  assumption,  this  equation 
has  a  root,  say  02.  Removing  from  /i(x)  the  factor  x  —  a-y,  we 
obtain  a  new  equation,  /2(x)=  0,  of  degree  n  -  2,  and  so  on. 
After  n  —  1  steps,  by  which  n  —  1  roots  are  removed,  we  have 
an  equation  of  the  first  degree  which  gives  one  more  root.  Hence 
there  are  just  n  roots. 

275.  To  Form  an  Equation  Having  Given  Roots.  —  Let  it  be 
required  to  form  an  equation  whose  roots  are  Oi,  a-z,  as,  .  .  .  an- 

Obviously  the  required  equation  is 

Ai,x-  ai)(x  -  a2){x  -  a^)  .  .  .  {x  -  aj  =  0, 
A  being  an  arbitrary  constant. 

Exercises.     Form  the  equations  whose  roots  are: 

1.  1,  2,  3.  3.   2,  2,  -2,  0.  5.    ±1,  h  i- 

2.  l!-l,  2.  4.    -1,-2,-3,-4.  6.    -h  h  i- 
(Write  the  results  from  exercises  5  and  6  with  integral  coefficients.) 


276,277]     ■  THEORY   OF   EQUATIONS  251 

276.  Relations  between  Coefficients  and  Roots.  —  In  the  case 
of  2,  3,  and  4  roots  respectively  we  find  on  expanding, 

(.r  —  ai){x  —  ao)  =  x^  —  (ai  +  02)  -k  +  ai«2. 
{x  -  ax){x-  a2){x  -  03)=  x^  -(oi  +  a-z -{- a^)  x- 

-{-{aiao -\- aoaz -\- a\a:i)x  —  aiaoaz- 
{x  -  ai){x  —  a2)(,x  -  as) (a;  -  a^)  =  x"^  -  (01+0-2  +  03  +  04)^3  + 

(•  •  •)  x^  —{■•■)  x  -\-  01020304. 

We  here  observe  three  facts,  namely : 

1.  The  coefficient  of  the  leading  term  is  unity; 

2.  The  coefficient  of  the  second  term  is  the  negative  sum  of  the  roots; 

3.  The  constant  term  is  the  product  of  the  roots,  plus  when  the 
number  of  roots  is  even,  minus  when  odd. 

We  shall  show  by  induction  that  these  results  are  true  in  general. 
Assume  them  to  be  true  for  n  —  1  roots;  then  if  the  equation 
whose  roots  are  Oi,  02,  .  .  .  o,j_i,  be 

a;'^-i+5ix"-2+  .  .  .  +g„_i  =0, 

we  have  by  hypothesis, 

9i  =  -(ai+«2+  •  •  •  +an-i);     gn-i  =  (-l)""^aia2  •  .  .  a„. 

Multiplying  the  above  equation  by  {x  —  a„),  which  introduces 
the  new  root  o„,  we  find  on  collecting  in  powers  of  x: 

x"  -{-(qi  -  a„)  a;"-i  +  •  •  •   -  o„9„_,  =  0, 
or,  X"  -(oi  4-  02  +  •  •  •   +  a„_i  +  a„)  x"-  1  +  .  .  . 

+  (-  l)"oia2   .  .  .  o„_iO„  =  0. 

Hence  if  the  results  are  true  for  the  case  of  n  —  1  roots,  they 
hold  also  for  n  roots.  But  they  are  true  for  4  roots,  hence  also 
for  5,  hence  for  6,  and  so  on. 

Exercise.  Show  by  induction  that  the  coefficient  of  the  third  highest  power 
of  X  equals  the  sum  of  the  products  of  the  roots  taken  two  at  a  time. 

277.  Fractional  Roots.  —  An  equation  with  integral  coefficients, 
in  ivhich  the  coefficient  of  the  leading  term  is  unity,  cannot  have  as 
a  root  a  rational  fraction  in  its  lowest  terms. 


252  THEORY  OF  EQUATIONS  [278 

Proof.     Assume  that  the  equation 

has  integral  coefficients  and  that  one  of  its  roots  is  7  '  where  a  and 
h  are  integers,  relatively  prime.     Putting  x  =  rwe  have, 

Multiplying  through  by  6""^  and  transposing, 

Here  we  have  a  fraction  in  its  lowest  terms  equal  to  an  integer, 
which  is  impossible.     Hence  y  cannot  be  a  root. 

Corollary.  Any  rational  root  of  an  equation  whose  coefficients  are 
integers  and  whose  leading  coefficient  is  unity  must  he  an  integer. 

278.  Imaginary  Roots.  —  //  the  general  equation  of  nth  degree, 
with  real  coefficients,  has  an  imaginary  root  a  +  ib,  then  also  the 
conjugate  imaginary,  a  —  ih,  is  a  root. 

Proof.     Assume  that  a  +  ih  is  a  root  of  the  equation 

f{x)=X''-\-piX''-'^-{-p2X''-^-\-    ■    •    •    +  p,^.iX  +  Pn=  0. 

Then 

(a  +  ib)^  +  Pi{a  +  ih)''-''  +  P2  (a  +  ih^-^ 
+  .  .  •  +p„_i(a  +  i6)+p,  =  0. 

Expanding  the  binomials,  reducing  all  powers  of  i  to  ±  1  or  ±  i, 
and  collecting  terms,  we  have  a  result  of  the  form 

f{a  +  ih)  =P-\-iQ  =  0. 

Hence  P  =  0     and    Q  =  0.     (258.) 

Now  substitute  a  —  ib  for  x  and  proceed  as  before.  The  result 
will  be 

/  (a  -  ih)  =P-  iQ, 

since  the  only  difference  is  in  the  sign  of  i.     But  P  =  0  and  Q  =  0, 
hence  P  —  iQ  =  0,   or  /(a  —  ib)  =  0.     Therefore  a  —  ib  is  a  root. 


279-281]  THEORY  OF  EQUATIONS  253 

■    We  may  state  our  result  as  follows:  Imaginary  roots,  if  present 
at  all,  always  occur  in  conjugate  pairs. 

279.  Multiple  Roots.  —  When  an  equation  has  two  or  more  roots 
equal  to  the  same  value  "  a,"  then  "  a  "  is  called  a  multiple  root. 

Suppose  that  the  equation 

f{x)=0 
has  m  roots,  each  equal  to  a.     Then 

f{x)  =  {x-arQ, 

where  Q  is  a  new  polynomial. 

Letf'ix)  denote  the  first  derivative  of  f  (x)  with  respect  to  x; 
then 

f(x)  =  (x-  a)-^  +  m  (x  -  ar-^Q. 

This  shows  that/'(x)  contains  the  factor  {x  -  a)'^''^,  and  hence 
that,  if  f{x)=0  contains  a  root  "a"  repeated  m  times,  f'{x)  =  0  will 
contain  this  root  repeated  m—  \  times;  f  {x)  andf'{x)  will  then  have 
the  factor  {x  —  a)"*~^  in  common. 

Hence  we  have  the  following  rule  for  finding  multiple  roots  of 
the  equation  /  {x)  =  0. 

Find  the  H.  C.  F.  (13)  of  f  {x)  andf'{x);  to  a  factor  {x  -  a^-"^  of 
the  H.  C.  F.  corresponds  a  factor  {x  —  a)*"  of  f{x). 

280.  Exercises.  Test  for  multiple  roots  and  find  all  the  roots 
of  the  equation.s: 


1. 

2-3  _  3  j2  -(-  4  =  0.                               6.   x4  -  3  x3  -  7  x2  +  1.5  X  +  18  =  0. 

2. 

j3  -  3  X  -  2  =  0.                                6.   x"  +  4  x3  -  16  x  -  16  =  0. 

3. 

x4-2x3-llx2+12x  +  36=0.      7.   x"  -  x3  -  3  x2  +  5  x  -  2  =  0. 

4. 

2.4-2  x3-39  x2  +  40x  +  400  =  0.      8.    4  x"  +  6  x3  +  5  x2  -  6  x  +  4  =  0. 

9. 

9  x4  -  54  x3  +  80  x2  +  6  X  -  9  =  0. 

LO. 

72  x5  -  276  x-i  +  278  x3  +  4.5  x2  -  108  x  -  27  =  0. 

281.  Transformation  of  Equations.  —  In  the  following  discus- 
sion we  assume  that  any  missing  powers  of  x  are  inserted,  supplied 
with  zero  coefficients,  so  as  to  make  the  equation  formally  com- 
plete.    We  consider  the  equation 

/  (x)   =  PqX^  +  Pl-T""^  +  p^X""--  +     •     •     •     +  Pn-lX  +  p„  =  0. 


254  THEORY  OF  EQUATIONS  [281 

I.  To  change  the  signs  of  the  roots. 

Put  X  =  —  y.     We  obtain, 

vq{-  yY+  vxi-  yY-'+V2{-  yT-'-+  •  •  •  +P«-i(-2/)+Pn=o, 

or,  on  multiplying  through  by  (—  1)", 

Por-Piy"-'  +  ?>2r-'-  •  •  •  +(-i)"-i?>n-iy+(-irPn=o. 

Hence,  to  change  the  signs  of  the  roots,  change  the  signs  of  alternate 
coefficients,  beginning  with  the  second  term. 

II.  To  multiply  the  roots  by  a  constant  factor,  m. 

Replace  xby—     (so  that  i/ =  mx). 

Then 

-(iT+-(r'+-(»r+ •••+--©+-=«• 

Multiplying  through  by  m",  we  have. 

Hence,  to  multiphj  the  roots  by  a  constant  factor  m,  multiply  the 
coefficients  in  order,  beginning  with  the  second,  by  w,  m^,  m^,  .  .  .  m". 
When  w  =  —  1  we  obtain  the  preceding  rule  for  changing  the 
signs  of  the  roots. 

III.  To  increase  the  roots  by  a  constant  quantity,  fi. 
Replace  xhy  y  —  h(so  that  y  =  x  -]-  h).     Then 

Po  (y  -  hr  +  Pi  (y  -  hr-'  +  P2  (y-hr-^+  ■  • . 
+  p„_i(t/-/o  +  p.  =  0. 

Expanding  the  binomials  and  collecting  in  powers  of  y,  we 
obtain  a  result  of  the  form, 

We  shall  now  show  how  to  obtain  the  coefficients  Pi,  Po,  •  •  •  Pn- 
Replacing  y  in  the  last  equation  by  x  -\-  h,  the  result  must  be 
the  original  equation,  /  (x)  =  0.     Hence 

fix)  =  po(x-{-hr  +  Pi{x  +  hr-'-\-P2(x  +  hr-^--\-  ■  • . 

+  P„_i(.r+/0+P.. 

This  shows  that  if  /  (x)  be  divided  hy  x -\- h,  the  remainder  is  P„. 
If  the  quotient  be  divided  hy  x  -\-  h,  the  remainder  is  P„- 1 ;  divid- 
ing the  second  quotient  by  {x  +  /i),  the  remainder  is  Pn-2,  and  so  on. 


282' 


THEORY  OF  EQUATIONS 


255 


Hence,  to  increase  the  roots  of  the  equation  by  h,  divide  f  (x)  by 
X  -\-  h,  then  divide  the  quotient  by  x  -\-  h,  divide  the  new  quotient  by 
X  +  h,  and  so  on.     The  successive  remainders  are,  in  order, 


Pji,  Pn-l,  Pn 


P\. 


A  concise  method  for  performing  the  required  divisions  will  be 
explained  in  the  next  article. 

282.  Synthetic  Division.  —  When  h  and  the  coefficients  po, 
Pi,  p2,  •  •  •  Pn  are  integers,  the  work  of  dividing  f  (x)  may  be 
performed  by  the  method  of  synthetic  division.  We  shall  illustrate 
this  by  increasing  the  roots  of  the  equation 


8x-15  =  0 


by  2. 


Performing  the  first  division  at  length,  we  have: 


x^+Ox^  -8x-  15 
x3  +  2a;2 


X  +2 

x'^  —  2x  —  4  quotient. 


-2x2 

-8a: 

-2^2 

-4x 

-4a;- 

15 

-4a;- 

8 

—    7  remainder. 

We  first  shorten  this  operation  by  omitting  to  write  the  powers 
of  X,  using  only  the  detached  coefficients,  thus: 


1  +  0 

1  +  2 

-2 

-2 


15 


15 


This  may  be  written  more  compactly  as  follows: 


1+0-8-15  I  +2 
+2-4-    8 


1st  quotient 


1-2-4 


7  remainder. 


256  THEORY  OF  EQUATIONS  [282 

Dividing  the  quotient  by  a;  +  2  we  have, 

1-2-4  I  +2 
+  2-8 
2nd  quotient  1-41  +  4  remainder. 

Dividing  the  second  quotient  by  x  +  2  we  have, 
1  -4  I  +2 
+  2 
3rd  quotient  1 1  -  6  remainder. 

The  whole  operation  may  now  be  written  thus: 
1  +  0-8-15  1  +2 


+  2 


1-2 

+  2 


7  1st  remainder 


1-4 

+  2 


+  4  2nd  remainder 


ij  —  6  3rd  remainder. 

Then  the  transformed  equation  is: 

a:3  -  6  a;2  +  4  X  -  7  =  0. 

To  diminish  the  roots  of  an  equation  by  h,  proceed  as  above 
with  X  —  h  in  place  of  x  -\-  h.  As  an  example,  we  diminish  by  4 
the  roots  of  the  equation 

a;4  -  5  a;3  +  7  a;2  -  17  a;  +  11  =  0. 

1-    5+    7-17  +  11  [-4 


-4+4 

-12  +  20 

1  -    1+    3-    5 
-    4-12-60 

—    9  1st  remainder 

1+    3  +  15 
-    4-28 

+  55 

2nd  remainder 

1+    7 
-    4 

+  43 

3rd  remainder 

1|+11 

Hence  the  transforr 

a;4  + 

4th  remainder. 

ned  equation  is: 

Ilx3  +  43.c2  +  55a;  -9  =  0. 

283,  284  ] 


THEORY   OF  EQUATIONS 


257 


In  using  the  method  of  synthetic  division  note  that  the  coeffi- 
cient of  the  leading  term  remains  unchanged. 
283.   The  graph  of  the  equation  y  =  /(x),  when 

f{oc)  =  p^Jc"-}-2>ix"-^-\-2>'i^"~^-^  '   '   '  +i>«-i-^+/>«. 

To  construct  the  graph  which  shall 
represent  the  fluctuating  values  of  y 
as  X  varies,  we  assume  a  series  of 
numerical  values  for  x,  calculate  the 
corresponding  values  of  y,  and  plot 
the  points  {x,  y).  On  drawing  a 
smooth  curve  through  these  points, 
we  obtain  a  graph  such  as  that  in 
the  figure,  which  represents  the  equa- 
tion 

y  =  x^  —  2x  —  1.  y  =  3?  - 

Here  a  set  of  corresponding  values  of  x  and  y  are : 

x=      0,        1,  2,  .  .  .  ,   -  1,   -  2,  . 


^ o l_X 

m 


2x 


y 


2,3, 


0, 


Since  the  curve  crosses  the  rc-axis  when  y  =  0,  we  see  that  the 
abscissas  of  the  points  where  the  graph  of  the  equation  y  =  f  (x) 
crosses  the  X-axis  {called  the  x-intercepts  of  the  graph)  are  the  real 
roots  of  the  equation  f  {x)=  0. 

An  inspection  of  the  above  graph  shows  that  one  root  of  the 
equation  a;^  —  2a:—  1=0  is  —1,  another  root  lies  between 
—  1  and  0,  and  the  third  between  +1  and  +2.  On  removing 
the  factor  x  +  1  from  this  equation,  the  depressed  equation  is 
x^  —  X  —  I  =  0.  Hence  the  exact  values  of  the  other  two  roots 
are  i  (l  ±  Vs),  or  approximateh',  +1.62  and  —0.62. 

284.  Effect  of  Changing  the  Constant  Term.  —  Suppose  that 
we  add  a  quantity  k  to  the  constant  term  of  /  {x)^  so  that  the 
equation 

y  =  f  (x) 

becomes  y  =  f  (^)  +  ^>-'- 

Suppose  the  curve  y  =  f  (x)  to  be  plotted ;  on  adding  k  to  each  of 
its  ordinates,  we  obtain  the  graph  of  y  =  f  (x)  +  k.     That  is,  if 


258 


THEORY  OF  EQUATIONS 


[284 


k  he  added  to  the  constant  term  of  the  equation  y  =f{x),  the  graph 
is  displaced  vertically  through  the  distance  k,  upward  if  k  is  plus, 
downward  if  k  is  minus. 
As  an  example,  consider  the  equations 


(1) 
(2) 
(3) 


1  ^3  _ 


-2a; +  2, 
-2a; +  4, 
-2x  +  6. 

The  graphs  are  shown  in  figure  (a).     The  curves  are  of  precisely 
(o)  the  same  form,  but  (2)  lies  two 

units  higher  than  (1),  and  (3) 
two  units  higher  than  (2). 


(6) 


VY 

f 

^ 

j 

\ 

r 

// 

^ 

\ 

/ 

// 

M 

\ 

/ 

r 

'/ / 

^ 

\ 

V 

^\ 

// 

\^ 

\^ 

f" 

// 

A 

\ 

Ji 

X 

/ 

\ 

\ 

/ 

\ 

J 

^ 

/ 

O/    \ 

I 

/  " 

\ 

o^ 

\ 

J 

X, 

o. 

A. 

Instead  of  displacing  the  curve 
vertically,  say  upward,  the  same 
effect  is  produced  in  the  graph 
by  moving  the  a:-axis  an  equal  distance  downward.  Thus  equa- 
tions (1),  (2),  (3)  are  represented  graphically  by  the  curve  in 
figure  (6),  y  being  counted  from  the  lines  OiZi,  O2X2,  O3X3 
respectively. 


285,286]  THEORY   OF  EQUATIONS  259 

285.  Occurrence  of  Imaginary  Roots  in  Pairs.  —  We  can  now 

consider  article  (277)  geometrically.     Thus  in  the  first  figure  of 
(284),  graph  (1)  shows  that  the  equation 

.  ^x-^  -x^  -2x-^2  =  0 

has  three  real  unequal  roots;  replacing  2  by  4,  the  two  positive 
roots  become  equal;  that  is,  the  equation 

ia;3_a^2_2a;  +  4  =  0 

has  three  real  roots,  two  of  which  are  equal;  finally  on  replacing 
4  by  6,  the  two  equal  roots  become  imaginary;  that  is,  the  equation 

Ix^  -x^  -2x  +  Q  =  0 

has  one  real  root  and  two  imaginary  roots. 

In  general,  by  changing  the  constant  term  in  f{x),  the  graph  of 
y  =  f{x)  may  be  raised  or  lowered  so  that  one  of  the  "elbows  " 
of  the  curve,  which  at  first  is  cut  by  the  x-axis,  will  become  tangent 
to  the  X-axis,  and  on  further  changing  the  constant  the  x-axis  will 
fail  to  intersect  this'  elbow.  Thus  two  real  unequal  roots  first 
become  equal,  then  imaginary. 

286.  Exercises.     Multiply  the  roots  of  the  equation 

1.  x3  +  .t2  -X  -  1  =0      by  2; 

2.  a;3  _2x  +  l  =0  by  -2; 

3.  x3  -48x  -  112  =  0      by  i; 

4.  x44-6x3  +  3x2 -26x -24  =  0      by  -J. 

Multiply  the  roots  of  the  following  equations  by  the  smallest  factor  which 
will  make  all  coefficients  integers 

6.     2;2  _|_  x  +  i   =  0.  8.     X3  -  .1  X2  +  .01  X  =  0. 

6.  \  x3  -  x2  +  3V  =0.  9.   x3  +  »  x2  -  ,\  =  0. 

7.  a;2  _  i  X  -  i  =  0.  10.   x4  4- 1.2  x2  -  .225  x  +  .015  =  0. 

Increase  the  roots  of  the  equation 

11.  x3  -  3  x2  +  4  =  0        by  2. 

12.  4  x3  -  3  X  -  1  =  0      by  3. 

13.  x4-2x3-llx2  +  12x  +  36  =0       by  -2. 

14.  x*  -  2  x3  -  39  x2  +  40  X  +  400  =  0     by  -4. 


260 


THEORY  OF  EQUATIONS 


■287 


In  the  following  equations  increase  the  roots  by  a  quantity  such  that  the 
term  involving  the  second  highest  power  of  x  shall  disappear. 


17.  x3  -  3  a:2  -  6  X  +  1  =  0. 

18.  x"  -  4  x3  -  8  X  +  32  =  0. 


15.  x3  -  3  x2  +  2  =  0. 

16.  x3  -  2  x2  +  1  =  0. 

In  the  following  equations  change  the  constant  so  that  two  roots  shall 
become  imaginary. 

19.   x3  -  x2  -  2  X  =  0.  21.   x3  -  3  X  -  2  =  0. 


3  x2  +  3  =  0. 


X  +  1  =  0. 


Solve  the  following  equations,  given  one  root. 

23.  x3 -2x2+x -2  =  0;       x=V^- 

24.  2  x4  -  3  x3  +  5  x2  -  6  X  +  2  =  0;       x  =  -  2  V^. 

25.  x5  -  8  x3  -  8  x2  +  64  =  0;       x  =  -  1  -  V^. 


y 

/ 

\ 

/ 

\ 

/ 

\ 

o 

\ 

\ 

\ 

\ 

/ 

\ 

/ 

V 

/ 

^ 

287.  Approximation  to  the  Roots 
of  an  Equation.  —  In  this  article 
we  shall  illustrate  a  method  for 
obtaining,  to  any  desired  degree  of 
accuracy,  any  real  root  of  an  alge- 
braic equation.  As  an  example  we 
shall  obtain,  to  four  decimal  places 
inclusive,  one  of  the  roots  of  the 
equation 

(1)       /(a;)=x3-  40^2  +  4  =  0. 

The  graph  is  given  in  the  figure. 

First  Step.  Location  of  Real 
Roots.  We  first  locate  the  real 
roots  by  trial.     As  a  set  of  corre- 


sponding values  of  x  and  /  (x)  we  have 

X  =-2,     -1,        0,    +1,    +2,    +3, 


20, 


1,    +4,    +1, 


+4. 
+4. 


f(x)  = 

When  /  (x)  changes  sign,  the  graph  crosses  the  a:-axis,  and  at  least 
one  root  must  lie  between  the  corresponding  values  of  x.  Hence 
there  is  a  root  between  —1  and  0,  another  between  -fl  and  +2, 
and  a  third  between  +3  and  +4.  But  there  cannot  be  more  than 
three  roots,  since  a  cubic  expression  cannot  contain  more  than 
three  linear  factors.  Hence  there  is  just  one  root  in  each  of  the 
above  intervals. 


287]  THEORY  OF  EQUATIONS  261 

We  shall  proceed  to  obtain  the  root  between  1  and  2. 
Second  Step.     Diminish  the  roots  of  the  given  equation  by  the  inte- 
gral part  of  the  root  required  (281). 

1 -4+0+4 [-1 
-1+3+3 


-3-3 
-1+2 


+  1 


1  -2 
-  1 


^ 


The  transformed  equation  is 

(2)  a;3  -  a;2  -  5  a:  +  1  =  0. 

Since  (1)  has  a  root  between  1  and  2,  (2)  must  have  a  root 
between  0  and  1,  that  is,  a  decimal  root.  To  make  this  root  an 
integer,  we  take  the 

Third  Step.  Multiply  the  roots  of  the  transformed  equation  by  10 
(281). 

The  new  equation  is 

(3)  x^  -  10  a:2  -  500  X  +  1000  =  0. 

The  root  of  (3)  between  0  and  10  will  give  the  first  decimal  of  the 
required  root  of  (1).  If  we  neglect  the  terms  in  x^  and  x-  in  (3) 
we  get  an  approximate  value,  x  =  2.  Putting  x  =  2  in  (3),  the 
left  member  is  negative;  now  putting  x  =  1,  the  left  member  is 
positive.  Hence  the  root  lies  between  1  and  2,  and  the  required 
root  of  (1)  is  1.1 +  . 

We  now  repeat  these  steps  and  obtain  the  first  decimal  of  the 
root  of  (3),  which  will  be  the  second  decimal  of  the  root  of  (1),  and 
so  on.  Indicating  the  three  steps  in  order  by  (a),  (6),  (c),  we 
obtain  the  successive  decimals  of  the  root  as  shown  below,  the 
process  of  finding  the  first  decimal  being  included  for  completeness. 

(3)  x3  -  10  x2  -  500  X  +  1000  =  0. 

(a)     Locate  the  root  between  0  and  10. 

Neglect  terms  in  x^  and  x^ ;  then  x  =  2.  Try  this  value  and  the 
next  smaller  value  (or  larger,  if  the  left  member  of  (3)  does  not 
change  sign)  and  the  root  is  located  between  1  and  2. 


262 


THEORY  OF  EQUATIONS 


[287 


(6)    Diminish  roots  by  figure  found  in  (a) . 
1  -  10  -  500  +  1000  |j 
-    1+      9+509 


1  -    9  -  509  +    491 

-1+8 

1-    8 

-    1 

-517 

Transformed  equation:  x^  —  7 x"^  —  517a;  +  491  =  0. 
(c)    Multiply  roots  hy  10. 

x3  -  70  a;  -  51,700  x  +  491,000  =  0. 
Repeat  these  operations  on  the  last  equation. 

(a)  re  =  491,000  4- 51,700  =  9+ . 

By  trial  the  sign  of  the  left  member  is  +  when  a:  is  9  and  8,  but 
changes  when  x  is  10.     Hence  the  root  is  between  9  and  10. 
(6)  1  -  70  -  51,700  +  491,000  |^ 

-    9+       549  +  470,241 


1  -  61  -  52,249 
-    9+       468 


+    30,759 


1-52 


53,717 


(c) 


lj-43 
x^  -  43a;2  -  52,717  a;  +  20,759  =  0. 
-  430  a;  -  5,271,700  a;  +  20,759,000  = 


0. 


The  required  root  of  (1)  is  now  x  =  1.19+.     Another  repetition 
of  the  process  gives  the  third  decimal. 

(a)  X  =  20,759,000  -=-  5,271,700  =4-. 

The  left  member  has  opposite  signs  for  a;  =  3  and  a;  =  4, 
the  root  is  between  3  and  4. 

(6)  1  -  430  -  5,271,700  +  20,759,000  |j-_3 

-      3  +         1,281  +  15,818,943 


hence 


1  -  427  -  5,272,981 
-      3+         1,272 


+    4,940,057 


424 
3 


5,374,353 


1|-431 
a;3  -~421  a;2  -  5,274,253  x  +  4,940,057  =  0. 
We  thus  have  the  required  root  of  (1)  as  a?  =  1.193+. 


287] 


THEORY  OF  EQUATIONS 


263 


We  may  omit  step  (c)  in  our  last  operation  and  get  the  next 
figure  of  the  required  root  by  neglecting  r'^  and  x-  in  the  last 
equation.     This  gives  x  =  .9+,  and  our  root  is,  finally, 
X  =  1.1939  + . 

A  convenient  arrangement  of  the  whole  operation  of  finding 
this  root  is  as  follows: 

1-4  +  0  +  4  |-  1 
-1+3+3 


3-3 

1  +  2 


+  1 


1-2 
-  1 


1  -  10  -  500  +  1000  |-  1 

-    1  +      9+509 


1  -    9  -  509  +    491 

-1+8 

1-    8 
-    1 

-  517 

u- 


1  -  70  -  51,700  +  491,000 

-    9+       549  +  470,241 


1  -  61  -  52,249 
-    9+       468 


+    30,759 


1  -  52  -  53,717 

-    9 


If 


1-43 


1  -  430  -  5,271,700  +  20,759,000  |  -  3 

-      3  +         1,281  +  15,818,943  ' 


1  -  427  -  5,272,981 
-      3  +         1,272 

+     4,940,057 

1-424 
-      3 

-  5,274,253 

1  -  421 

Root,   1.193 

9+. 

264  CUBIC   EQUATIONS  [288-290 

288.  In  approximating  to  the  roots  of  an  equation,  the  fol- 
lowing remarks  should  be  borne  in  mind.  Let  the  student  supply- 
proofs  when  needed. 

(1)  Every  equation  of  odd  degree  has  at  least  one  real  root. 
(For  / (x)  has  opposite  signs  when  x  =  -\-'X  and  x  =  —  oo.) 

(2)  When  an  even  number  of  roots  lie  between  x  =  a  and  x  =  b, 
f  (a)  and  /  (b)  will  have  like  signs. 

(3)  Whenever  more  than  one  root  lies  between  two  assumed 
values  of  x,  especial  care  must  be  used  to  separate  them  by  trial. 

(4)  The  next  decimal  of  a  root  is  obtained  approximately  by 
dividing  the  absolute  term  of  the  last  transformed  equation  by 
the  coefficient  of  x  with  its  sign  changed. 

(5)  Should  this  decimal  be  too  large,  the  constant  term  of  the 
next  transformed  equation  will  change  sign.  (Observe  that  in 
the  example  the  constant  terms  of  the  original  equation  and  of 
all  the  transformed  equations  are  of  the  same  sign.) 

(6)  Should  this  decimal  be  too  small,  the  next  transformed 
equation  will  not  have  a  root  between  0  and  10,  except  when  there 
happen  to  be  two  or  more  roots  of  the  original  equation  with  the 
same  integral  part. 

(7)  To  obtain  a  negative  root,  change  the  signs  of  all  the  roots 
and  proceed  as  for  a  positive  root. 

289.  Exercises.  Calculate  to  four  decimal  places  the  real  roots 
of  the  equations: 

1.  x3  -  24  X  -  48  =  0.  12.   4  x3  -  3  X  -  1  =  0. 

2.  x3  -  7  x2  +  4  X  +  24  =  0.  13.   x"  +  x3  -  2  x2  -  3  x  -  3  =  0. 

3.  x3  -  2  X  +  1  =  0.  14.   x4  -  2  x3  -  8x2  +  24x  -  48  =  o. 

4.  x3  -  x2  +  X  -  1  =  0.  15.   x4  -  4  x3  -  8  X  +  32  =  0. 

5.  x3  +  x2  +  X  +  1  =  0.  16.   x4  +  2  x3  +  X  +  2  =  0. 

6.  x4- 6x2 +  5=0.  17.   3x4-2x3 -16x2-56x  +  96  =  0. 

7.  x3  -  7  X  -  5  =  0.  18.   x3  -  7  X  -  7  =  0  . 

8.  x3  -  31  X  -  19  =  0.  19.    8x4  +  16  x3  +  18  x2  +  X  +  7  =  0. 

9.  x3  -  48  X  -  112  =  0.  20.    7  x3  +  8  x2  -  14  X  -  16  =  0. 

10.  2  x3  -  18  x2  +  46  X  -  30  =  0.      21.   2  x*  -  5  x3  -  32  x  +  80  =  0. 

11.  7  x3  -  9  X  +  5  =  0.  22.   2  x5  -  4  x3  +  3  x2  -  6  =  0. 

290.  Cardan's  Solution  of  the  Cubic  Equation.  —  As  in  the  case 
of  the  quadratic  equation,  so  the  equations  of  third  and  fourth 


290]  CUBIC   EQUATIONS  265 

degree  may  be  solved  by  means  of  radicals.  This  cannot  be  done 
for  equations  of  degree  higher  than  the  fourth.  We  give  here  a 
solution  of  the  cubic  equation 

(1)  aox3  +  3  aia;2  +  3  a2X  +  ag  =  0. 

We  first  obtain  a  new  equation  containing  no  term  of  second 
degree.     To  do  this,  put 

X  =  ij  -{-  h. 

Expanding  and  collecting  in  powers  of  y, 

aoy^  +  3  (ao/i  +  aO  if  +  3  (00/1^+  2  a^h  +  02)  2/  +  aoh^ 
+  3  aih-  +  3  02/1  +  03  =  0. 

The  term  in  y-  drops  out  if 

aoh  +  ai=0,     or     h= ^• 

With  this  value  of  h  the  equation  becomes 

o   ,  3  (0002  —  ar)       ,   ao^as  —  3  aoaia2  +  2  ai^ 
"''^   + ^ ^  + a? =^- 

Putting  y  =  —' 

we  have 

2^+3  (aotta  -  ai2)  z  +  (ao^ag  -  3  aoaia2  +  2  ai^)  =  0. 
Let 

H  =  0002  -  fli^;    G^  =  floras  -  3  ao«ia2  +  2  Oi^. 
Then  the  equation  becomes 

(2)  z^-\-3Hz-\-G  =  0. 
To  solve  this  equation  let 

2  =    Vr  +  Vs. 

Then 

23  =  r  +  s  +  3  \/rs  (Vr  +  -s/s), 

or,  2^  -  3  's/rs  •  2  -  (r  +  .s)  =  0. 

If  this  is  to  be  identical  with  (2),  we  must  have 

yfrs  =  —  H,     and    r  +  s  =  —  G; 
or,  rs  =  —  H^,    and     r  -{-  s  ^  —  G. 


266  .  CUBIC   EQUATIONS  [290 

Solving  for  r  and  s, 

-G+  \/W+nP  -G-  VG2  +  4 ^3 

'  = 2- '    '=  2 

Then 

z  =  ^r-^  <fs=<Jr-^'     {rs=-m.) 
\lr 

Let  the  three  cube  roots  of  r  be  ai,  0:2,  and  0:3.  Then  the  three 
values  of  z  are 

H.  H  H 

Zi  =  ai ,    22  =  «2 '    23  =  0:3 

(Xi  a2  as 

The  corresponding  values  of  x  are  then  found  from 

,   ,  ai       z       ax      z  —  a\ 

x  =  y-{-h  =  y = i  = — i- 

ao      ao      flo  Go 

Nature  of  the  Roots.  —  The  following  criteria  serve  to  deter- 
mine the  nature  of  the  roots : 

(a)  G^  +  4H^  <  0,  three  real  distinct  roots; 

(b)  G^  +  4iH^  =  0,  three  real  roots,  two  being  equal; 

(c)  G^  +  4  H^  >  0,  one  real  root,  two  imaginary  roots. 
By  direct  calculation,  for  which  we  shall  not  take  space,  we  find 

{z,  -  z-i)  {Z2  -  23)  (23  -  21)  =  V-27((?2  +  4/^3), 
or, 

{zi  -  Z2Y  {Z2  -  zzY  {zs  -  z,Y  =  -  27  (G2  +  4  H^).        , 

When  the  roots  are  all  real,  their  differences  are  real,  hence  the 
left  member  of  the  last  equation  is  positive;  therefore  G^  +  4  H^ 
must  be  negative.  When  two  roots  are  eqYial,  their  difference  is 
zero;  hence  G^  +  4  H^  =  0.  When  two  roots  are  imaginary,  they 
must  be  conjugate  imaginaries;  suppose  them  to  be 
Z\  =  a  +  ih     and     Z2  =  a  —  ib. 

Let  the  third  root  be  23  =  c,  where  c  is  real  [(1),  (288)].  Then  we 
show  directly  that  (z^  —22)^  is  negative,  and  that  (22—23)^(23—21)^ 
is  positive,  hence  the  left  member  of  the  above  equation  is  nega- 
tive; therefore  G^  -{-  4:  H^  must  be  positive. 

The  quantity  G^  -\-  4  H^  is  called  the  discriminant  of  the  cubic 

z^ -j- Z  Hz -\- G  =  0, 


291]  QUARTIC   EQUATIONS  267 

When  all  the  roots  are  real,  i.e.,  G--\-  4  H^  <0,r  and  s  are  con- 
jugate complex  quantities;  let  them  be 

r  =  A  -\-  iB;  s  =  A  —  iB. 

In  this  case  Vr  and  '^ s  cannot  be  evaluated  algebraically.     The 
roots  may  then  be  obtained  in  trigonometric  form.     Let 

A  =  u  CO?,  v;    B  =  u  sin  v. 
Then 

r  =  u  (cos  V  -\-i&m.v);  s  =  u  (cos  v  —  i  sin  v). 

Hence 

Vr  =  Vm(cos ^ h  *sm ~ 1, 

/       v-\-2kT!r       .   .    v-\-2kTr\     ,       „   ,    „ 
Vs  =  Vw  (  cos ^ I  sm ^ J ;  A;  =  0,  1,  2. 

Here  ^u  denotes  the  real  cube  root  of  u. 
We  now  find 

z=</-r^</-s^2</u  cos  'L±1J^  ;k  =  0,l,2. 

o 

291.   Ferrari's  Solution  of  the  Quartic  Equation.  —  Write  the 
given  quartic  equation  in  the  form 

(1)  x^ -\- 2  ax^  +  bx-  -\- 2  ex -\- d  =  0. 
Add  to  both  members  (px  -\-  q)^: 

(2)  x'^-j-2ax^-}-{b  +  p-)x^+2(c  +  pq)x+(d  +  q^)  =  {px  +  qy. 

The  left  member  will  become  a  perfect  trinomial  square  of  the 
form 

.    (a;2  -{-  ax  -\-  k)^  ^ 

by  putting 

(3)  p^  =  a'^-b-\-2k;    q^=-d-\-k^;    pq  =  -c^-ak. 
Then  equation  (2)  becomes 

(a:2  -\-ax  +  kY  =  {px  +  qY, 
or, 

(4)  x^-\-ax+k=±{px  +  q). 

Taking  each  sign  in  turn  we   have  two  quadratic   equations 
in  x,  which  give  the  four  roots  of  (1). 


268  QUARTIC  EQUATIONS  [291 

To  obtain  the  values  of  p,  q,  and  k  in  (4)  we  must  solve  equa- 
tions (3)  for  these  quantities  in  terms  of  the  coefficients.  On 
equating  the  values  of  p^q"^  from  the  product  of  the  first  two  of 
equations  (3)  and  the  square  of  the  third  equation  we  find  a 
cubic  to  determine  A;: 

(5)         2  F  -  6A;2  +  2  (ac  -d)k-{-(hd-  a?d  -  c")  =  0. 

This  is  called  the  reducing  cubic,  and  is  to  be  solved  for  a  real 
value  of  k.     Then  p  and  q  are  obtained  from  (3). 

Example.  x*  +  4  x3  -  3  a;2  -  16  x  +  5  =  0. 

Here  a  =  2,   6  =  -  3,   c  =  -  8,   d  =  5. 

Then  (5)  is  2  A;3  +  3  A:2  -  42  A;  -  99  =  0. 

A  real  root  is  k  =  —  3. 

Then  from  (3),  p  =  1,   q  =  2;     or,     p  =  -  1,    5  =  -2. 

With  either  set  of  values  of  p  and  q  (4)  becomes 

(x2  +  2x-3)  =  ±(x  +  2). 


Hence 

Therefore 


x2  +  X  -  5  =  0,     or,     x2  +  3  X  -  1  =  0. 
-1±V21  -3±Vl3 


X  = 


2 


Exercises.     Solve  the  following  equations: 

1.  x3  -  3  x2  +  4  =  0.  9.   x4  +  2  x3  +  2  x2  -  2  X  -  3  =  0. 

2.  x3  -  3  X  -  2  =  0.  10.   x*  +  6  x3  +  3  x2  -  2  X  -  3  =  0. 

3.  4  x3  -  3  X  -  1  =  0.  11.   x4  -  4  x3  -  9  x2  +  2  X  +  3  =  0. 

4.  x3  -  24  X  -  48  =  0.  12.   x^  +  4  x3  -  16  x  +  H  =0. 

5.  x3  -  7  x2  +  4  X  +  24  =  0.  13.   x"  +  4  x3  -  16  x  -  16  =0. 

6.  x3  -  3  x2  -  6  X  +  1  =  0.  14.   x4  -  3  x3  -  7  x2  +  15  X  +  IS  =  0. 

7.  x3  -  7  X  -  6  =  0.  15.   x4  -  4  x3  -  8  X  +  32  =  0. 

8.  x3  -  x2  +  X  -  1  =  0.  16.   x*  -f  x3  -  2  x2  -  3  X  -  3  =  0. 


CHAPTER  XX 

Spherical   Trigonometry 

292.  Spherical  Geometry.  —  We  devote  this  article  to  a  review 
of  some  facts  concerning  the  geometry  of  the  sphere. 

(a)  A  plane  section  of  a  sphere  is  a  circle.  When  the  plane 
passes  through  the  center,  the  section  is  a  great  circle;  otherwise  a 
small  circle. 

(b)  Any  two  great  circles  intersect  in  two  diametrically  opposite 
points  and  bisect  each  other. 

(c)  The  two  points  on  the  sphere  each  equally  distant  from  all 
the  points  of  a  circle  on  the  sphere  are  called  the  'poles  of  the 
circle.     A  great  circle  is  90°  distant  from  each  of  its  poles. 

(d)  A  spherical  triafigle  is  a  figure  bounded  by  three  circular 
arcs  on  a  sphere.  In  this  chapter  we  consider  only  triangles  whose 
sides  are  arcs  of  great  circles.  Any  such  triangle  may  therefore 
be  considered  as  cut  from  the  spherical  surface  by  the  faces  of  a 
triedral  angle  whose  vertex  is  at  the  center.  The  face  angles  of 
this  triedral  angle  measure  the  sides  of  the  triangle,  and  its  diedral 
angles  the  angles  of  the  triangle. 

(e)  If  a  triangle  be  constructed  by  striking  arcs  with  the  vertices 
of  a  given  triangle  as  poles,  the  nev/  triangle  is  called  the  polar 
triangle  of  the  given  one. 

Let  the  sides  of  the  given  triangle  be  a,b,  c;  its  angles  A,  B,  C; 
let  the  sides  of  the  polar  triangle  be  a',  b',  c'  and  its  angles  A',  B' ,  C; 
we  assume  that  A  is  the  pole  of  a',  B  of  b',  and  C  of  c';  then 

a'  =  180-  A  ;  A'  =  180  -  a  ; 

and  similarly  for  the  other  sides  and  angles.  That  is,  ariTj  part  of 
the  polar  triangle  is  the  supplement  of  the  part  opposite  in  the  given 
triangle. 

(f)  The  difference  between  the  sum  of  the  angles  of  a  spherical 
triangle  and  180°  is  called  its  spherical  excess. 

The  area  of  a  spherical  triangle  is  to  the  area  of  the  sphere  as  its 
spherical  excess,  in  degrees,  is  to  720°.     That  is,  if  E  be  the  spherical 

269 


270 


SPHERICAL  RIGHT  TRIANGLES 


[293,294 


excess  in  degrees  and  K  the  area,  and  R  the  radius  of  the  sphere, 
then 

293.  Spherical  Right  Triangles.  —  Let  0  be  the  center  of  a 
sphere  and  ABC  a  triangle  on  its  surface  having  C  =  90°.  The 
triangle  shown  in  the  figure  has  each 
part,  except  C,  less  than  90°.  The  re- 
sults below  are  true  in  any  case,  as  may 
be  shown  by  drawing  other  figures,  or  by 
assuming  the  right  triangle  as  a  special 
case  of  the  oblique  triangle. 

Cut  the  triedral  angle  0-ABC  by  a 
plane  _L  OB,  forming  the  plane  right  A 
A'B'C,  with  C'=90°.  Then  also  As  OB'C  and  OB' A'  are 
right-angled  at  B'.  Further,  Z  A'B'C  measures  Z  B  (292,  (d)). 
Then 

A'C 

•     AiT^i^t      ^'C      OA'      sin& 
(a)  sin  B  =  sm  A'B  C   =  -^Tg,  = = 


(b)  cos  B  =  cos  A'B'C  = 


(c)  tan^  =  tsin  A'B'C  = 


B'C 
A'B' 


A'C 
B'C 


A'B' 
OA' 

sine 

B'C 

OB' 
A'B' 

tana 
tan  c 

OB' 

A'C 
OC 

tan  6 

B'C 

OC 

sin  a 

Dividing  (a)  by  (b)  and  comparing  with  (c)  we  have 
(d)  cos  c  =  cos  a  cos  b. 

By  combining  these  equations  we  may  obtain  others  by  which 
any  part  of  the  triangle  may  be  expressed  directly  in  terms  of 
any  two  given  parts,  the  right  angle  excluded.  These  formulas 
are  all  contained  in  two  simple  rules. 

294.  Napier's  Rules  of  Circular  Parts.  —  Let  co-a:  denote  the 
complement  of  any  part  x  of  the  triangle.     Take  the  complements 


295] 


SPHERICAL  RIGHT  TRIANGLES 


271 


of  c,  A,  B,  and  arrange  the  five  parts,  a,  b,  co-A,  co-c,  co-B,  called 
circular  parts  in  the  order  in  which  they  occur  in  the  triangle  as 
in  the  adjacent  figures.  Then  if  any  one  of  the  five  be  taken  as 
the  middle  part,  of  the  other  four  parts  two  will  be  adjacent  and 


rco-;^ 


fMB^ 


the  other  two  opposite  to  this  part.     Thus,  if  co-c  be  taken  as  the 
middle  part,  co-B  and  co-A  are  adjacent,  a  and  b  opposite. 
Rules: 

i  Product  of  tangents  of  adjacent  parts, 
or 
Product  of  cosines  of  opposite  parts. 

Exercise.  Taking  each  part  in  turn  as  the  middle  part  write  out  a  com- 
plete hst  of  formulas  relating  to  the  spherical  right  triangle.  Derive  these 
formulas  from  those  given  above. 


295.   Solution  of  Right  Triangles. 

Example.     Given  a  =  35°  42';  5  =  60°  25' 
The  diagram  of  circular  parts  is  shown  in 

the  figure.     Taking  (1),  (2),  (3)  in  turn  as 

middle  part  we  have 


Find  b,  c,  A. 


(1) 
(2) 
(3) 

Hence, 


sin  35°  42'  =  tan  29°  35'  tan  b; 
sin  29°  35'  =  tan  35°  42'  tan  (co-c); 
sin  (co-A)    =  cos  29°  35'  cos  35°  42'. 


^  _  sin  35°  42'  _  sin  29°  35' 

^^"^  ^  "  tan  29°  35"  ^"^^  "^  ~  tan  35°  42" 
cos  A  =  cos  29°  35'  cos  35°  42'. 

Check.     The  computed  parts  must  satisfy  the  relation 
sin  (co-A)  =  tan  b  tan  (co-c),  or  cos  A  =  tan  b  cot  c. 


272 


SPHERICAL  OBLIQUE  TRIANGLES 


[296,  297 


Computations. 

log 
sin  35°  42'  =  9.7660 
tan  29°  35'  =  9.7541 


log 
sin  29°  35'  =  9.6934 
tan  35°  42'  =  9.8564 


log 
cos  29°  35'  =  9.9394 
cos  35°  42'  =  9.9096 


tan&  =0.0119 
b  =  45°  17 
Check. 


cot  c  =  9.8370 
c  =  55°  30' 
log  cos  A  =  log  tan  b  +  log  cot  c. 
9.8490  =  0.0119  +  9.8370. 


cos  A  =  9.8490 
A  =  45°  4' 


Ambiguous  Case.  When  the  given 
parts  are  an  angle  (not  the  right  angle) 
and  its  opposite  side,  two  solutions 
are  possible,  because  the  other  parts 
are  then  calculated  from  their  sines. 
The  two  triangles  together  form  a  lune,  as  A  A'  in  the  figure, 
where  A,  a  are  supposed  to  be  the  given  parts. 

296.  Quadrantal  Triangles.  —  A  quadrantal  triangle  is  one 
having  a  side  equal  to  a  quadrant  or  90°.  Its  polar  triangle  will 
be  a  right  triangle,  which  may  be  solved  by  Napier's  Rules.  The 
parts  of  the  given  quadrantal  triangle  then  become  known  by 
(e)  of  (292). 

Solve  the  following  triangles,  C  being  the  right  angle: 
4.   6  =  100°,  7.   B  =  145°  53', 

a  =    40°.  c  =  110°  20'. 


Exercises. 

1.  a  =  45°  10', 
B  =  70°  20'. 

2.  6  =  65°  15', 
A  =  25°  50'. 

3.  c  =  33°  18', 
b  =  30°  37'. 


A  =  120°  42' 
c  =  56°  50'. 


5.  A 


Solve  the  following  quadrantal  triangles: 
10.   a  =  90°,  11.   A  =  65°  15', 

b  =  50°,  b  =  90°, 

c  =  40°.  c  =  50°  25'. 


8. 

b  =   132°  16', 

B  =    65°  46'. 

9. 

c   =  170°  4', 

a  =  175°  17'. 

12. 

A   =  122°  10', 

B  =  70°  22', 

c  =  90°. 

We 


297.   Oblique  Triangles.     Two  Fundamental  Formulas. 

consider  only  triangles  in  which  no  part  exceeds  180°. 

I.   Law  of  Sines.  —  Let  ABC  he  Si  spherical  triangle.     Draw 
CD  ±  AB,  forming  two  right  triangles  (figure). 

In  A  ACD,   sin  p  =  sin  &  sin  A. 

In  A  BCD,   sin  p  =  sin  a  sin  B. 


J 


298]  SPHERICAL  OBLIQUE  TRIANGLES  273 

Therefore,  c 

sin  h  sin  A  =  sin  a  sin  B,  or 
-      sin  a  _  sin^i 
sin  6  ~  sin  B 

That  is,  the  sines  of  the  sides  are 
proportional  to  the  sines  of  the 
opposite  angles. 

Exercise.     Discuss  the  case  in  which  D  falls  on  AB  produced. 

II.  Law  of  Cosines.  —  In  the  figure  above  let  AD  =  m,  so  that 
BD  =  c-  m.     Then  in  right  A  BCD 

cos  a  =  cos  (c  —  wi)  cos  p,  .  .  .  (d),  (293) 
=  cos  c  cos  m  cos  p  +  sin  c  sin  m  cos  p. 

But  in  A  A  CD 

cos  W  cos  p  =  COS  6 

and  sin  m  cos  p  =  sin  C  sin  h  X  — — 7^  =  sin  b  cos  A. 

^  sinC 

Hence 

(2)  cos  a  =  cos  h  cos  c  +  sin  6  sin  c  cos  A. 

That  is,  the  cosine  of  any  side  equals  the  product  of  the  cosines  of 
the  other  two  sides  plus  the  product  of  their  sines  by  the  cosine  of 
their  included  angle. 

Exercise.    Discuss  the  case  where  D  falls  on  AB  produced. 

From  the  fundamental  formulas  (1)  and  (2)  we  shall  derive  a 
series  of  other  formulas  adapted  to  the  solution  of  triangles. 

298.  Principle  of  Duality.  —  By  means  of  (e)  of  (292)  any 
formula  relating  to  the  spherical  triangle  can  be  made  to  yield  a 
second  formula.  Thus,  let  A  A'B'C  be  polar  to  A  ABC.  Then 
from  (1)  and  (2) 

sin  a'      sin  ^4'  ,  ,,  /   ,     •    r/   •      r         ai 

—. — Ti  — -■ — ^^;       cos  a  =  cos  0  cose  +smosmccosA. 
sm  6       sm  B 

But  a'=180-yl,       .4' =  180  -  a, 

6'  =  180  -  B',      B'  =  180  -  6, 
c'  =  180  -  C,      C  =  180  -  c. 


274  SPHERICAL  OBLIQUE  TRIANGLES  [299 

Substituting  and  reducing,  we  have 

sin  A  _  sin  a 
sin  B       sin  6 
(3)  cos  ^  =  —  cos  B  cos  C  +  sin  -B  sin  C  cos  a. 

The  first  of  these  is  simply  the  law  of  sines;  the  second  is  a  new 
formula. 

299.  Formulas  for  the  Half  Angle.  —  Solving  (2)  for  cos  A,  we 
have 

cos  a  —  COS  6  cos  c 


cos  A 


sin  h  sin  c 


Then 


sm^ 


1  .      ,  /l  -  cos  A      /„,,         ,    ,   .  Il  —  cosA^\ 
2^  =  V 2 (,Whynot±\/ ^ V 


_  cos  g  —  cos  6  cos  c 
sin  b  sin  c 


sin  b  sin  c  —  cos  a  +  cos  b  cos  c 


2  sin  b  sin  c 


-W'- 


_  .  /cos  {b  —  c)—  cos  a 
V        2  sin  6  sin  c 


_.     a  +  6  —  c.    a  —  6  +  c 
2  sm ;r sm ?: 


2  sin  6  sin  c 
Now  let 

(4)  2s  =  a  +  6  +  c; 

then 

a  +  6  —  c                       ,     a  —  b-\-  c  , 
^ =  s  —  c     and     ^ =  s  —  b; 

therefore, 

,_.  -1^4  /sin  (.<*  —  b)  sm  (s  —  c) 

(5)  sm-^  =  V ^ — .    ;    .     -• 

^  ^  2  V  sm  &  sm  c 

Similarly, 


/c^  1^4  /sm.s  sm  (.s  —  a) 

(6)  cos-^l  =  V .    ,    . ^' 

^  ^  3  V        smfrsmc 

By  dividing 
(7) 


2  >       sm  ft-  sm  {s  —  a) 


300,301]  SPHERICAL  OBLIQUE  TRIANGLES  275 

Given  the  three  sides,  one  of  these  formulas,  preferably  the  last, 
will  determine  the  angles.     When  all  three  angles  are  desired,  let 


(8)  tan  r 

then 


/sin  (s  —  a)  sin  (s  —  b)  sin  {s  —  c) , 


Sins 


,_,  ,      1   ^  tanr 

(9)  tan-^1  =  -:— 7 r 

^  ^  2  sin  (s  —  a) 

/,.^^  .      1  _  tanr 

(10)  tan-l?=-^^ -Ti 

^     ^  2  sin  (s  —  b) 

/,,N  .^      1  ^  tanr 

(11)  tan-  C  = 


2  sin  {s  —  c) 

300.  Formulas  for  the  Half  Sides.  —  Proceeding  as  above  with 
(3)  of  (298),  or  by  applying  the  principle  of  duality  to  formulas 
(5)  to  (11)  we  have,  on  putting 

(12)  2S  =  A  +  B-\-C 

and  ' 


(13)  tan  Il  =  \/-  ''''^''^ 


cos  {S  —  A)  cos  {S  —  B)  cos  {S  —  €) 


(14) 
(15) 


—  cos  S  cos  {S  —  A) 
sin  B  sin  € 


1  ,  /cos  is  -  B)  cos  (,S'  -  C) 


sin  B  sin  C 


,_,         ^1  ./     -  cos -S  cos  (-S' -  ^) 

(17)  tan|«  =  tani?cos(«  -  ^), 

(18)  tan|6  =  tan  JJ  cos  (*S'  -  jB), 

(19)  tan  I  c  =  tan  72  cos  {S  -  C). 

301.   Napier's  Analogies. —  Dividing  tan|A   by  tan  |  5  and 
reducing,  we  have 

tan  I  ^  _  sin  (s  —  h) 
tan  I B      sin  (s  —  a) 

By  composition  and  division, 

tan  I A  +  tan  |  B  _  sin  (.s  —  6)  +  sin  (s  —  a)  _ 
tan  I A  —  tan  |  B  ""  sin  (s  —  6)  —  sin  (s  —  a) 


276  SPHERICAL  OBLIQUE  TRIANGLES  [301 

Reducing  tangents  to  sines  and  cosines  and  simplifying  the  result- 
ing complex  fraction,  applying  the  formulas  for  sin  {x  ±  y)  on  the 
left  and  for  sin  u  ±  sin  t;  on  the  right,  we  have 

sin  Ty{A-\-  B)  tan  |  c 


(20) 
or, 


sini  (^  —  B)      tan|  {a  —  b) 


1  ,  ,       sin  5  (^  -  -B)  ^      1 

(200  tanj  («  -  6)  =  ^-^^-^i^-c. 

Multiplying  tan  h  A  by  tan  h  B  and  reducing, 

tan  ^  A  tan  ^  B  _  sm(s  —  c) 
1  sin  c 

By  composition  and  division,  and  reduction  as  above, 

cos  I  (^  +  ^)  _       tan  I  c 
^^^^  cos  1{A-  B)  ~  tan|(«  +  6)' 

or, 

^      1  ^      ,      .       cos  i  (A  -  B)        1 
(210  tan-(.  +  6)=^^^^^^^^^tan-c. 

These  formulas  determine  the  other  two  sides  when  two  angles 
and  their  included  side  are  given. 

Proceeding  as  above  with  tan  ^  a  and  tan  ^  b,  or  by  the  principle 
of  duality  applied  to  formulas  (20)  to  (21'),  we  obtain 

sin  I  {a  -\-b)  _       cot|  C 
^^^^  sin lia-b)  ~  tan r,{A-B)' 


or, 


1  ,       sin :;  (a  —  b)        i 

(220  tan-(^-^)  =  ^.^;^^^_^^^cot-C, 


(23) 
or, 


cos  k  («  +  b)  cot  I  (7 


cos  I  (a  -  fe)      tan  i  (^  +  B) 


1  cos  i  (a  —  b)        1 

(230  t^°i(-^  +  ^)%osi(«  +  >)'°'^^- 

These  formulas  determine  the  other  two  angles  when  two  sides 
and  their  included  angle  are  given. 


302,303]  SPHERICAL  OBLIQUE  TRIANGLES  277 

302.   Area  of  a  Spherical  Triangle.  —  This  may  be  calculated  by 

(f)  of  (312),  namely. 

K  =  ^  ^'!rg^^'^  X  4  TT  72^    or,    K  =  E  (radians)  X  R". 

To  obtain  E,  we  may  first  calculate  the  angles.     E  may  also  be 

obtained  by  one  of  the  following  formulas,  which  we  add  without 

proofs. 

1  tan  5  a  tan  r,  b  sin  C 

tan-E 


3  1  +  tan  r,  a  tan  r,  h  cos  C ' 


tan-^  =  y  tan- tan — - — tan — - — tan  • 

303.  Solution  of  Spherical  Oblique  Triangles.  —  Six  cases 
arise,  according  to  the  nature  of  the  three  given  parts. 

I.  Given  two  sides  and  an  opposite  angle. 

Denote  the  given  parts  by  a,  b,  A.  Calculate  B  by  (1),  then 
C  by  (22)  or  (23),  and  c  by  (20)  or  (21). 

„,     ,  sin  b       sin  B 

Check:  -. —  =  -. — rz> 

sm  c      sm  C 

which  involves  the  computed  parts. 

Ambiguous  Case.  Formula  (1)  will  give  two  (supplementary) 
values  for  B.  Two  solutions  are  obtained  when  both  values  of 
B  lead  to  values  of  C.  Otherwise  one  or  both  values  of  B  must 
be  rejected. 

Rule.  Retain  values  of  B  which  make  A  —  B  and  a  —  b  of  like 
sign.     Otherwise  (20)  and  (22)  take  the  impossible  form  +  =  —  • 

II.  Given  two  angles  and  an  opposite  side. 

Denote  the  given  parts  by  ^,  B,  a.  Calculate  b  by  (1),  then 
proceed  as  in  I. 

Ambiguous  Case.  Formula  (1)  gives  two  values  of  b.  Retain 
the  value  or  values  ivhich  make  A  —  B  and  a  —  b  of  like  sigii. 

III.  Given  the  three  sides. 
Calculate  the  angles  by  (9),  (10),  (11). 

^,     ,  sin  A      sin  B      sin  C 

Check: =  -. — j-  = 

sm  a       sm  o       sm  c 


278  SPHERICAL  OBLIQUE  TRIANGLES  [304 

IV.  Given  the  three  angles. 

Calculate  the  sides  by  (17),  (18),  (19). 
Check:  As  in  III. 

V.  Given  two  sides  and  their  included  angle. 

Denote  the  given  parts  by  a,  h,  C.  Calculate  ^  {A  -\-  B)  by  (23'), 
^  {A  —  B)  by  (22');  then  A  and  B  by  addition  and  subtraction; 
obtain  c  by  the  law  of  sines.     Check  by  (20)  or  (21). 

VI.  Given  two  angles  and  their  included  side. 

Denote  the  given  parts  by  ^,  B,  c.  Calculate  |  (a  +  6)  from 
(21'),  I  (a  -  6)  from  (20') ;  hence  get  a  and  b;  obtain  C  by  the  law 
of  sines.     Check  by  (22)  or  (23). 

304.   Example.     Given  a  =  100°  37',  h  =  62°  25',  A  =  120°  48'. 

Formulas. 

.    „      sin  6  .     . 
sm  B  =  - —  sm  A , 
sm  a 

^  .  ^      sin  I  (a  +  fo)  ^      ,  ,  .       D\ 
cot  I  C  =  -^-7 — —~  tan  i(A  -  B), 
^  sm  I  (a  —  6) 

.  sin  I  (A  +  B)  ,       .  ,        , . 


^,     ,                       sin  6      sin  B 
^^''^'                     sinc'sinC 

Computations. 

log  sin  b  =  9.9476                    a  =  100°  37' 

A  =120°  48' 

log  sinyl=  9.9340                     b=    62°  25' 

B=    50°  46' 

colog  sin  a  =  0.0075             a  +  6  =  162°  62' 

A-\-B  =  170°  94' 

log  sin  j5  =  9.8891             a-b=    38°  12' 

A  -  5  =    70°   2' 

B=  50°46'.5    Ha  +  &)=    81°  31' 

^{A-\-B)=    85°  47' 

or      129°  13^5    |(a-&)=     19°    6' 

HA -5)=    35°    1' 

Reject  the  larger  value  of  B  by  the  rule  in  I. 

log  tan  i  (A  -  5)  =  9.8455  log  tan  ^  (a  -  b)  =  9.5395 

log  sin  i  (a  +  6)  =  9.9952  log  sin  ^  (A  ■{- B)  =  9.9989 

colog  sin  1  (a  -  6)  =  0.4852  colog  sin  ^  {A  -  B)  =  0.2412 

log  cot  ^  C  =  0.3259  log  tan  |  c  =  9.7796 

iC=    64°43'.5  ic  =  31°3' 

C  =  129°  27'  c  =  62°  6' 

Check:   log  sin  b  =  9.9476  log  sin  B  =  9.8891 

sin  c  =  9.9463  sin  C  =  9.8877 

0.0013  0.0014 


305,306]  TERRESTRIAL  SPHERE  279 

Note.  In  the  solutions  of  triangles,  a  complete  form  should  he  pre- 
pared in  advance,  so  that  only  numerical  values  need  be  inserted 
when  the  tables  are  opened. 

305.   Exercises.     Solve  the  triangles  whose  given  parts  are: 


1. 

a  =  53°  18'.3, 
b  =  36°  5'.6, 
c  =  50°  24'.9. 

2. 

a  =  42°  15'.3, 
h  =    33°  18'.8, 
c=  60°32'.l. 

3. 

a  =  84°  14'  30", 
b  =  44°  13'  46", 
c  =  51°  6' 20". 

4. 
.1  =  116°  8'.5, 
B  =  35°  46'.6, 
C  =  46°  33'.7. 

5. 

A=  97°  53', 
5=  67°59'.7, 
C=  84°46'.7. 

6. 

A   =  53°  42'  34", 
B=    62°  24' 26", 
C  =  155°  43' 12" 

7. 

a  =  89°  0', 
,  &  =  47°  30', 
.  C  =  36°  0'. 

8. 

a  =  70°  20', 
b  =  38°  28', 
C  =  52°  30'. 

9. 

b  =  19°  24', 
c  =  41°  36', 
A  =  84°  10'. 

10. 

a=    88°  24'  3", 
c  =  120°  10'  55", 
B  =  49°  27'  50". 

11. 

a   =  102°  22', 

B  =    84°  30', 

,  C  =  125°  28'. 

12. 

h  =    76°  40'  48' 
A=    84°  30' 20' 
C  =  130°  51'  33' 

13. 

c  =  104°  13'.4, 
A   =  63°  48'.6, 
B=    51°46'.2. 

14. 

c  =  108°  39'  10", 
A=    64°  48' 52", 
B  =  40°  23'  17". 

15. 

,  6  =  54°  18'  16", 
A  =  127°  22'  7" 
C  =  72°  26'  40" 

16. 

a  =  88°  27'  50' 
,  b  =  107°  19'  52' 
.  C  =  116°  15'  0' 

17. 

b  =  83°  5'  36", 
c  =  64°  3'  20", 
A  =  57°  50'  0". 

18. 

b  =  68°  45', 
B  =  58°  5', 
C  =  50°  51'. 

19. 

a  =  56°  37', 
A  =  123°  54', 
B  =  57°  47'. 

20. 

a  =  48°, 
b  =  67°, 
A  =  42°. 

21. 

6=  81°, 
A=    72°, 
B  =  119°. 

22. 

a  =  69°  34'. 9, 
c  =  70°  20'.3, 
C  =  50°  30'.  1. 

23. 

a  =  69°ll'.8, 
b  =  56°  3S'.5, 
A  =  68°  40'. 

24. 

a  =  151°  01'  5" 
b  =  134°  10'  52" 
A  =  144°  20' 45" 

25. 

a=  40°  8' 28", 
b  =  118°  20'  8", 
A=    29°  45' 32", 

26. 

,  a  =  88°  12'.3, 
,  A=    63°15'.2, 
,  B  =  132°  18'. 

27. 

c  =  100°  49'  30", 
B  =  95°  38'  11", 
C=    97°  26' 28". 

28. 

A  =  45°, 
a  =  10°, 
b  =  60°. 

306.  Applications  to  the  Terrestrial  Sphere.  —  We  shall  con- 
sider the  earth  as  a  sphere  with  a  radius  of  3960  miles.  Longi- 
tudes are  to  be  reckoned  from  Greenwich  westward  through 
360°  or  24'^.     We  shall  denote  longitude  by  X,  latitude  by  0. 

Problem  1.  Given  the  latitudes  and  longitudes  of  two  stations, 
to  find  the  distance  between  them. 

Let  P  be  the  earth's  north  pole,  G  Greenwich,  Ai  and  Ao  the 
two  stations  (figure).  Let  the  positions  of  the  two  stations  be 
Xi,  </>!  and  X2,  02  respectively. 


280 


CELESTIAL  SPHERE 


[307 


Then 


in     AA1PA2,    PAi 
P 


90 


'  -cj^i,  PA2  =  90° -9^2,  and 
Z  A1PA2  =  X2  —  Xi-  Hence 
in  AA1PA2  two  sides  and 
their  included  angle  are  known, 
and  A1A2  (in  degrees)  maybe 
calculated  as  in  V  of  (303). 

Problem  2.  A  ship  is  to  sail 
from  A I  to  A  2  by  the  shortest 
path  (great  circle).  On  what 
course  (at  what  angle  with  the 
meridian)  will  she  depart  from 
^1;  on  what  course  will  she 


Assuming  the  positions  of 
A I  and  A2  given,  we  have  two 
sides  and  the  included  angle  of  the  triangle  A1PA2.  We  must 
calculate  angles  Ai  and  A2.     This  comes  under  V  of  (303). 

Exercises. 

1.   Calculate  the  sides  (in  miles),  the  angles,  and  the  area  (in  square  miles) 
of  the  triangle  whose  vertices  are: 


New  York 

-I  =  4  55  54, 

<;& 

=  40°  45'  N. 

San  Francisco 

8    9  43, 

37°  47'  N. 

Mexico  City 

6  36  27, 

19°  26' N. 

2.  A  vessel  sails  on  a  great  circle  from  San  Francisco,  ^  =  S*"  9°  43', 
^=  37°  47'  N.  to  Sydney,  /I  =  13"  55"  10',  (J)  =  33°  52'  S.  Find  the  courses  of 
departure  and  arrival  and  the  distance  sailed. 

3.  If  the  vessel  in  exercise  2  makes  12  knots  an  hour,  what  is  her  position 
(>l  and  4>)  and  on  what  course  is  she  sailing  5  days  after  leaving  San  Fran- 
cisco?    (1  knot  =  1  nautical  mile  =  1'  on  a  great  circle.) 

307.  Applications  to  the  Celestial  Sphere.  —  For  the  purpose 
of  this  article  we  assume  the  celestial  sphere  to  be  an  indefinitely 
large  sphere  concentric  with  that  of  the  earth.  On  it  as  a  back- 
ground we  see  all  celestial  objects. 

The  projections  on  the  celestial  sphere  of  the  earth's  poles, 
equator,  meridians  and  parallels  of  latitude  are  named  respectively 
the  celestial  poles  (P,  P'  in  the  figure),  the  celestial  equator  or 
simply  equator  (QwQ'e),  hour  circles  (as  PSE),  and  parallels  of 
declination  (as  MSM'). 


307] 


CELESTIAL  SPHERE 


281 


An  observer  at  0  on  the  earth's  surface  will  have  his  zenith  at 

Z,  where  the  plumb  line  at  0,  if  produced,  would  meet  the  celestial 

sphere;  his  horizon  is  the 

great   circle    sivne,   whose 

pole  is  Z;  his  meridian  is 

the   great    circle    nPZQs, 

meeting    the    horizon    in 

the      north      and     south 

points. 

Let  >S  be  a  point  on  the 

celestial    sphere,    as    the 

sun's    center,    or    a    star. 

Because    of    the    rotation 

of  the  earth,  *S  will  appear 

to    describe    the    parallel 

e'MSw'M'e',   rising   at  e' 

and  setting  at  w'.    When 

S  has  the  position  shown  in  the  figure,  HS  is  its  altitude,  denoted 

by  h  (height  above  horizon) ;  Z  sZH  (measured  by  arc  sH)  is  its 

azimuth,  denoted  by  A;  ZS,  or  90°  -  h,  is  the  zenith  distance  of  .S 

and  denoted  by  z.     Thus  h  and  ^,  or  z  and  A,  completely  define 

the  position  of  S  with  reference  to  horizon  and  zenith. 

With  reference  to  the  equator  and  pole, 
^*S  is  called  the  declination  of  S,  denoted 
by  d,  and  Z  QPE  (angle  which  hour 
circle  PS  of  S  makes  with  meridian  PQ) 
is  called  its  hour  angle,  denoted  by  t;  PS 
or  90°  —  0  is  the  polar  distance  of  S,  and 

denoted  by  p.     Thus  the  position  of  S  is  defined  by  3  and  t,  or  by 

p  and  t. 

A  PZS  is  called  the  astronomical  triangle;  its  parts,  except  the 

angle  at  S  which  we  shall  not  need,  are: 


PZ  =  90°  -  nP  =  90° 
PS  ^  p  =  90°  -  8; 
Z  ZPS  =  t; 


c/);       (</j  =  latitude  of  0.) 
ZS  =  z  =  90°  -  h; 
Z  PZS  =  180°  -  A. 


Problem  \.     Given  the  latitude  of  0,  and  the  declination  and 
altitude  of  S,  calculate  the  hour  angle  and  azimuth  of  S. 


282 


CELESTIAL  SPHERE 


[307 


Here  the  three  sides  of  A  PZS  are  known,  and  it  is  only  neces- 
sary to  calculate  the  angles  at  P  and  Z  (III,  303). 

Problem  2.  In  a  given  latitude,  and  for  a  given  declination  of 
the  sun,  find  the  sun's  hour  angle  at  sunset  and  the  length  of  day 
(sunrise  to  sunset). 

Here  S  is  on  the  horizon  and  PZS  a  quadrantal  triangle.  We 
obtain  t  by  solving  the  polar  right  triangle  for  180  —  t.  The  length 
of  day  will  be  2  t. 

Problem  3.  Given  the  sun's  declination  and  its  hour  angle 
when  it  bears  due  west  (A  =  90°),  find  the  latitude. 

Here  PZS  is  a  right  triangle,  with  the  right  angle  at  Z;  p  and  t 
are  known,  and  PZ  may  be  calculated  by  use  of  Napier's  Rules. 

Problem  4.  Find  the 
hour  angle  and  azimuth  of 
Polaris  when  at  greatest 
elongation,  given  the  dec- 
lination of  the  star  and  the 
latitude  of  the  station  of 
observation. 

Let  MSM'  be  the  star's 
diurnal  path  about  the  pole 
(figure).  When  the  star  is 
at  greatest  elongation,  the 
great  circle  ZS  is  tangent  to  the  small  circle  MSM',  of  which  PS 
is  a  radius.  Hence  A  PZS  is  right-angled  at  S;  PZ  and  PS  are 
known,  and  the  angles  at  P  and  Z  may  be  found  by  aid  of  Napier's 
Rules. 


Exercises. 

1.  In  latitude  40°  49'  the  sun's  altitude  is  observed  to  be  20°  20';  its 
declination  is  15°  12';  find  its  azimuth  and  hour  angle. 

2.  With  latitude  and  declination  as  in  exercise  1,  find  the  sun's  hour  angle 
when  it  is  due  west;  when  it  sets;  find  its  azimuth  at  sunset;  find  the  length 
of  day. 

3.  With  latitude  and  declination  as  in  exercise  1,  find  the  sun's  altitude 
and  azimuth  when  its  hour  angle  is  45°. 

4.  The  sun,  in  declination  12°  22',  is  observed  to  have  an  altitude  of  30° 
when  due  west.     What  is  the  latitude  of  the  station? 

5.  The  declination  of  Polaris  being  88°  49',  find  his  azimuth  and  hour  angle 
at  greatest  elongation  at  a  station  in  latitude  40°  49'. 


307]  CELESTIAL  SPHERE  283 

6.  As  in  exercise  5  for  the  star  51  Cephei,  d  =  87°  11',  and  for  d  Ursae 
Minoris,  3  =  86°  37'. 

7.  The  stylus  of  a  horizontal  sundial  consists  of  a  rod  pointing  to  the 
north  celestial  pole.  Hence  its  shadow  falls  due  north  when  the  sun  is  on  the 
meridian,  that  is,  at  apparent  noon.  What  angle  does  its  shadow  make  with 
the  meridian  one  hour  after  apparent  noon,  at  a  place  in  latitude  40°? 

(Suggestion.  In  the  first  figure  of  this  article  let  nP  =  40°  and  Z  ZPS  = 
l*"  or  15°.  The  stylus  lies  in  the  line  P'P,  and  its  shadow,  cast  by  the  sun  S, 
must  lie  in  the  plane  SP'P,  and  hence  will  fall  on  the  plane  of  the  dial,  sivne, 
along  the  line  of  intersection  of  these  two  planes.  This  line  will  be  deter- 
mined by  the  center  of  the  sphere  and  the  point  where  arc  SP  produced  will 
meet  arc  7ie.  Call  this  point  S'.  Then  arc  nS'  measures  the  required  angle, 
and  maybe  found  by  solving  right  A  nPiS',  in  which  nP  =  40°  and  ZnPS'  =  15°). 

8.  What  angle  does  the  shadow  of  a  horizontal  sundial  make  with  its 
noon  position  t  hours  after  noon  in  latitude  (j)  ?  {Ans.  tan  x  =  tan  t  sin  0, 
X  being  the  required  angle.) 

9.  Calculate  the  angles  which  the  hour  lines  of  a  horizontal  sundial  make 
with  the  noon-line  in  an  assumed  latitude. 


ANSWERS 

(Answers  are  given  only  for  the  odd-numbered  exercises.) 


Article  10 


1.  ia-i.  3.  .05  a?  -Sab-  4.625  ac.  5.  63  x  -  2  y  -  4  2.  7.  a'^b'^c  - 
I  a2c4  +  1*3  a^cd^  -  2  a?c.  9.  1.2  a^bcW^  -  1.8  acM''  +  .3  a?cH^  -  3  (mW^.  '  11. 
x6-5x4  +  3a;3  +  6a;2-7x+2.  13.  x^  -  9  x^i/S  +  7  x^s -j_  13  a;3y4  _  19  a;22/5  4. 
8xy^-yT.     15.    |  a2  +  ,27.     17.   x^  -  a2.r2  -  62^2  +  02^2.     19.   9a2_9o+6. 

21.    -3«3p.    23.    -IS...    26.   ■°^'-"-J^'  +  ».    27.-i^  +  ||  + 

j2^-2-^-  29.  3a2x-4ax2+x3.  31.  .c2  +  5  x^/ +  3  ?/2.  33.  fa3-fa26 
+  |a62.  35.  a;3_  33.2  _2x-t-i.  37.  2x2 -fxy  -  ByS.  39.  (a  +  b)^.  41. 
Jx2-32/.    43.  ab  +  c.    45.  a2 +  62 +c2 +  d2  _2(a6 -oc  +  oti -6d  +  6c +c(f). 


Article  12 

^'  ^^~4l"2~^)-  3.  (x-l)(3x-l).  6.  (3x-?/)(2x  +  72/). 
7.  X  (2  X  -  3  ?/)  (4  x2  +  6  XT/  +  9  y^).  9.  (x  +  2)  (x  -  2)  (x  +  3)  (x  -  3). 
11.  (x  -  11)  (x  +  10).  13.  (x  -  9  a2)  (x  -  o2).  15.  {xy  -  5  z)  (xy  +  2  z). 
17.  (x  -  1)  (x  -  8)  (x  +  S).  19.  (x  +  1)  (x2  -  X  +  1)  (x  -  1)  (x2  +  x  +  1). 
21.  -  3  xy  (x  +  y).  23.  (ac  +  b)  (ac  +  d).  25.  xy  (x  +  y)  (x  -  y)2. 
27.    (x2y  -  22)  (x2y  +  5).     29.    (x  +  2)  (.t2  +  7^  +  2). 

Article  15 

1.  3  (x  +  1).  3.  4  (x2  +  7/2).  5.  ox  (a  -  x)2.  7.  3  a  (2  a  +  3  6  -  4  r). 
9.  (2x-3y).  11.  (3x-2g).  13.  (x2  +  7).  15.(5x2-1).  17.  (x  +  y). 
19.  (a2  -  ab  +  62).  21.  24  a26x2y3.  23.  (a  +  6)  (a  -  6)2.  25.  (x  -  4) 
(x+l)(x+3).  27.  (3x-2)  (2x+3)(2x-3).  29.  (3x  -  2a)  (4  x  -  3«) 
(3  X  +  4  a).  31.  (m  +  v)  (m  -  n)  (m  +  2n)  (m  -  2  n).  33.  (x  +  1)  (x  +  2) 
(X  +  3).  35.  (x  -  1)  (X  +  1)  (X  +  2)  (X  +  3).  37.  (x  -  1)  (x  +  1)  (.r2  +  1) 
(x2  -  X  +  1).     39.  (a  -  6)  (a  +  6)  (a2  +  ab  +  62)  (a^  -  ab  +  62)  (a^  +  a^b^  +  ¥). 

284 


ANSWERS  285 


Article  19 


1    4ax      3.  ""  ~  ^'      6.  x8+a:6y2+xV  +  xV+j/«      ^    ^      ^    x^+x  +  1 
'   X  —  y         '  x*y*  '     '       '      x2  + 1 

11.,-^^.   13.  ^(^±M.    15.-^4.   17.  x3+,3.     19.^-^-.    21.   0. 

3{x  —  y)  a2  m2  -^  4x  — ?/+3 

o,    2  -(3x4-2a:3  +  3a;-4)  x3  _  x2 

(x  -  1)  (x  -  2)  (x  -  3)  ■  2  (x  +  1)  (x2  +  1)  (x2  -  X  +  1)  a3      a^ 

+  ^_"^  7x/         343 x3\  (5 a3  - 9 c3)  (45 c3  -  49  63)  (9 c2  -  5 gS) 

^x2      u,-3'  II2/V         13312/3/    ^^-  14,175a3c6 


Article  21 


6.  ^„.    7.  ai269.     9.  '^^'- 


Article  33 

1-  «o^-   '•  ^^^^^'^  »•  £■  ^-  iJ;^-  »•  ^5^V4.   11.  -Vie, 

'"^  '  ^^~~  '25 

49' 
10 


727^13.    'm  m.  -^27.  .16.    ^1.    y/^|.    y/j||,.     17.    ^|^ 

V^w'  \/n'  "•  ■^^  ^'''  ■^'"-  "•  "^'  '^"''  '^°^"-  '''■  ^^ 
\/i'-  \/^°  ^^-  75'  "^^''  ~^^-  ^'- '  ^'^^  ''■ » "^^- 

31.  3^  -s/o.  33.  (3  +  6  -  a)  V«.  35.  a  \lx.  37.  1  +  V3.  39.  |  (\^  + 
Ve).  41.  I  (V6  -  Vl4).  43.  V2  +  V5."  45.  VTO  -  Vs.  47.  1. 
49.  6  V2  -  3  V 15  +  8  V3  -  6  VlO.     51.  8-8  <JVl  +  -s/ls.     53.  Vw2  -  n. 


z 
32      —  3V2" 


55.  a.     57.    ^/r32.v22.     59.  2.     61.  4.     63.  3.     65.  3.     67.    W| 

71.    7n'\     73.    a'.     75.    al      77.    a'\      79.    (x  +  2/)"*".      81.   ^-     83.    a*. 

„  a(l+^^)  Q_  4a3  +  12V^3  4-9  II+2V14  .,  a^h-^c\fd. 
^^-        l-a      ■    ^^'  4^^^:^9  ^®'  ■         5 ^^-     a26  -  c^d 

6 

93.  al    95.  2a-2-7a?  +  6a-^+7a-*-lla-^-2a-*  +  7a-i-6.    97.  2 a^ 

4  3  2  1 

-5a^  +  lOa^  ~7a^  +  6a^.  99.  4  a-'^b'^  -  12  a-h''^  +  9 a'^b-'^  101. 
X'  -3x2/^ +  3  X* 2/^ -2/2.  103.  m'^l  +  4  m"^  +  6m-3  +  4  rre"?  +  m-e). 
105.  a^  +  J  +  a^  +  2(a"^-a'^"-J^).  107.  a*  +  46^ +9c  +  16d^  +  2( -20*6* 
+3aic^-4aid^-6  6*c*  +  8  6*d^-12c*d0.  111.  x^  -  x^  +  x^  -  x^  +  1.  113. 
ah  +  a"  6"  +  6^.    115.  a  -\/a.    117.  3  ^f^;  5  ^T^;  9  V^.     119.  3  yfi. 


286  ANSWERS 

121.  2  </i.  123.  9  V^n;.  125.  m  V^.  127.  47  -  i.  129.  4  i  \/6  -  2. 
131.-1.  133.  i.  135.  ^-^+i?^^-  137.  -  i  139.  2  x  =  3.  141. 
ax  +  b  =  C.     143.  4x  ^  5.     145.  x  =  5.     147.  x  =  10.     149.  x  =  4. 

Article  38 

1.  0.     3.    -  3,  -  4,  -  6,  -  7.     5.   -  3,  -  4.     7.  2,  a.    9.    7,  .3.     13.  (p  +  q) 

Article  41 
3.   2.9196,  0.9196,  9.9196  -  10,  8.9196  -  10.     6.  3.667.     7.  1.655;  11.695. 
9.52.22)29.34.   11.0.1829.   13.  log  g-    15.  log  ^^^    17.  log"' ^^^ 

Article  46 
1.  0.975.     3.  88.444.     5.  0.99965. 

Article  51 

.           5a  —  36-           c     ,   m  —  -p     „            .          6           ^.         mn 
1.  X  = ^ 3.  X  =  T-   5.  -■   7.  00.     9.  r 7-   11.  — j » 

2  0  q—n  b  —  a  —  1  m  +  n—a 

00.  13.  1.     15.  |. 

Article  60 

■    1.  6.     3.  -^.     5.  -^.     7.  3tol.     9.  II  days.     11.     ,     "^ ^ , 
m  —  1  n  —  m  ao  -\-  ac  -{-  be 

days.     13.  5x\  min.  past  10;  21/i  min.  past  10.     15.    1^  hrs. 

Article  64 

1.  3,  5.     3.  32,  -  17.     5.  9,  8.     7.  2,  3.     9.  Inconsistent.     11.  0,  4.     13. 

1,  -  1.     16.  Dependent.     17.  6,  12.     19.  12,  5. 

Article  69 

1007.28  92.33 


1.  6,  12.     3.  6,  12.     5.  9,  7.     7.  4,  3. 


1.0163725'  "       1.0163725 
5  a  —  5  & 


2 


11.  Not  independent.    13.  5,  6.    15.  i  \.    17.  4,  7.    19.  7,  f .    21 

a  +  b        „-       abcg  abcf  __        _  nqrt -\- npsv  _      _qsl—  msqv        __ 

■ — ;r —       Zi>   -, u'    , :;■       «o.  X  —  ■ ■. ;  y  — ; •       At, 

2  bg  —  af     bg  —  af  mqr  +  ps  ps  +  mqr 

_r>fq^^  _nmq_^  29.  No  solution.  31.  No  solution.  33.20,17,5.  35. 
mq  —  np    mq  —  np 

3,2,1.  37.  3,4,5.  39.  i,  |,  oo.  41.  1,2,3,4.  43.  1,  .8,  .2,  .6.  45.  16t\  hrs-, 
7Hhrs.  47.  S4000;  4i%.  49.36,9.  51.89,35.  53.13,17,20.  55.  1,  If ,  li. 
57.  2,  3,  6  hrs.    59.  $9150,  $8600,  $7550. 


ANSWERS 


287 


Article  75 

1.  2,  -  6.      3.  2,  -  8.      6.  -  2,  7.      7.  3,  -  4f.      9.  5,  Ij 
13.   -2^,5.     15.  3,  -4^.     17.  2  6,  -6.     19.  2,  c. 


11.  t,  -^. 


Article  86 
1.6.    3.  0or3.    5.1.    7.13.    9.4.    11.  _Vl441-29     ^^^  ^      15.  3or-? 


25.  3.     27.   ±\/-mp.     29, 


17.  li     19.  15.     21.  ±  i  VI-     23.  4  or  -  1 

±6V«2^^fc2.     31.   ±ia.    33.  ±mV^.    35.  ^^'^^^,   ,    "  •   37.  b±\fW^^ 


39.  a 


y/^r 


41.  ±  8  or  ± 


a     /^ 
2-V^ 


a  +  1 

43.  ^^±^v^r:r4. 


46 


4  ^4  or  -  8.     47.  4  or  -  9.     49.  27  or  64.     51.  0  or  9.     53.  14,  16,  18;  or, 

b  -  a±Va2+62-6a6 


■14,  -16,  -18.     55.  30X60.    57.  ^  \/ab  -  A 
10 


61.  I     63.  . 

571 

-9;  a;<  -1  { 


,  n<l     65.  20;  60.     67.  x  >  -  1  and  <  -  9;  -  1  and 
>  -9.     71.  i(Vl7-l);  ^(Vl7  +  l). 

Article  93 


1.  ±  W2;  ±  I  ^/2.    3.  I  V2;  -  ^  V2.    5. 


-6±  VTi.3±2  vn 


or  If;  3  or  U-     9. 


6  ±  V6.  -  2  ±  3  V6 


20 


20 


11.  m  =±2. 


7.  0 


Article  95 

1.  ±4=;  t4=.    3.  -^-:-±..    5.  0,  2;  1,  0.     7.    l^^  j:  3^^1559 
Vl3         Vl3  Vl3    Vl3  145 


-162±2  V-1559      g    4^ 
145  ■ 


23 


13 


9±  V-23 
13 


11.   ±  I  V5- 


Article  97 


1.  0,  1;0,  1. 


«    65±Vl29    l±Vl29      _    7±4V^^   2T4 
o.  t:?^ ; 77; —  •    5. 7^ : 


32 


16 


1±V5;^^^-    9.   -4±2V3;  -7±4  V3.     11.   -  1. 


288  ANSWERS 

Article  99 
1.  1,-1;  0,-1.     3.  W5;  h     5. 


35  '  35 

54±V66     -.12T3V66 


25  25 


).  ±  V  v"^  ±  f  V-  7.   11.  db  Vs. 


Article  105 


1.   X  =  ±  I  \^,  ±1  VS,     y  =  ±h  V2,   +  W5.      3.   X  =  0,  3,  ±  r%  Vl3; 
7/  =  2,  0,  T  A  Vi3.     5.  x  =  0,  9,  V;  2/=0, -6,  V- 

Article  106 


1.   ±1  V29±V41,  ±  W7TV41.     3.  ±x«5  V-5;  ±V-¥.     5.  ±V3, 
±  j\  V57,  0,  T  t\  V57. 

Article  107 

1.  ±V3;±1.    3.±f;±|.    5.±^;±i^'. 


Article  111 

l.x=±25;y=±6.  3.±5;±4.  5.  i^;  ^-^-  7.  f,  i;  i,  |. 
9.  7^  499.  g^  923|_  11.  13-  7_  13.  ^  13;  ^  7;  two  solutions.  15.  7,  - 1; 
-  3,  17i  17.  37rt,  4;  43t\,  7.  19.  4,  5;  4,  3;  two  answers.  21.  14fi,  5; 
15H.  2.     23.  8,  9;  9,  8.     25.  ±2,  <x;  ±1,  o).     27.  ^^^^^^ ;  ^^^^ ;  four 

solutions.  29.  3  ±  V6,  ~^^^^~^^  ;  3  T  VS,  Zli^^/EU.  31.  _  2,  ^; 
0,  00.  33.  7,  2;  2,  7.  35.  0,  5;  5,  0.  37.  5,  -  6;  11,  -  12;  four  answers. 
39.  2,3,  -3±V3;  3,  2,  -3tV3.     41.  12,3,  -8±2V7;  3,12,  -8T2V7- 

43.  ,8,  ?,    -^'t^^;  8,54,^«^l:^.    46.  4, 3/ ^^5^;  3,4, 
3  ^  ^  ^ 

l^^Zm.    47.  4,7,ll-±:^p^;7,4,li^^^.    49.  9, 7;  7,  9;  two 

answers.     51.   ±  2,  ±  t?z  V516;  ±  1  ^  -7^=-     53.  x  =  7,   -  2,  77r,   -  2  t/?, 

V516  ^         I — T^ 

7u;2,  -2vfi;  2/ =  2,  -7,  2w,  -  7  ti',  2w',  -Tifi;  w=~ — 2~  '  ^^' 
m  =  11,  -9,  11  w,  -9w,  11^2,  -9it)2;  «  =  9,  -11,  9w,  -11  w,  9w^,  -llu;2. 


ANSWERS  .         289 

67.  X  =  243,  32;  ij  =  64,  729;  two  answers.     69.  x  =  3,  -  1,  +  1,  -  3;  ?/  =  1, 

-  3,  3,  -  1.      61.   ±  Vl  ±  \  V3;    ±  Vl  T  W3.      Use  both  upper  or  both 
lower  signs  under  radicals;    outside  of  radicals  use  all   combinations.     63. 

^^ 2 » 2 '    ^^^  solutions.     65.  ^ ; 


fcg  T  g  V2  62  -  a2 

2 


,  ...  -_      m    /      ^      n-\-ahm    ,   A         m 

two    solutions.         67.    k~  \±\    -^nr  +  1 J  !      ^7" 

2  a  v      y  n  —  3  a6w        /        2  a 

f  ±  \/-^  _  o"  r l);  two  solutions.     69.    ic  =  ±  V-5  +  1,  y  =  ±V^ -Ij 

x  =  ±V^  +  l,  y  =  ±V^-l;foursoIutions.  71.  14  =  ± W±32 V2-27+1; 
t;  =  ±  5  V  ±  32  V2  —  27  —  1 ;  four  solutions.  Use  all  possible  combinations  of 
signs  in  u  and  in  v.     73.  ±  ah\j2a'^  —  62  —  a62  ;  ±  ah'sj'la^  —  62  +  a62;  two 

solutions.      75.         ^        ;         '^  77.  5,  3,  4  ±  \r^;  3,  5,  4  T  V^^^. 

79.  z  =  ±  V^^,  2/  =  =F  V^^3,  z  =  2;  two  solutions.  81.  x  =  2  or  00;  y  = 
-  ^   or  —  1;    2  =  1  or  0.      83.   x  =  ^r-  (pg  -  r  ±  \J{pq  -  r)2  -  4  §3) ;  1/  ==  — - 

(pg  -  r  TV(pg  -J-P  -4^3);  2  =  -  ;  two  solutions.  85.  ±  V',  "F  V,  ±  ¥ ;  take 

all  upper  or  all  lower  signs.  87.  x  =  ^(a  —  6 — c— 2±  VC-^ +6+c  —  a)2  +4;a  (2+c)) ; 

I 

or  ±  J 


rT^''^rTx'    ®^'  ^  =  '*«^3;   2/=±iV^,or±f;2=±W-i, 


)^ 


Problems 
1.  8,6.  3.  48,  36.  5.  x  =  15,  -  12;  ?/  =  ll,  -  16;  two  answers.  7.  x=  19, 
■  20;  2/  =  17,  -18;  four  answers.  9.  33,  56.  11.  19,23.  13.  28, 20  ft.  sec. 
15.  13j"j;  45  days.  Assume  each  man's  pay  proportional  to  amount  of  work  he 
does.  17.42.  19.  ^5.  21.3,5yds.  23.  si  =  15.4;  11.7  ft.  sec;  S2  =  6.8;  12.2 
ft.  sec. 

Article  114 
1.  3.     3.   -  3.     5.  0.     7.-3.     9.  1.     11.   -  4.     13.   -  |.     15.  |.     17. 

i^l.     19.   ^'^'^^.    21.  1,-3.     23.    -  V.     25.   -6. 

log  -i'  log  u-36- 

Article  122 


26.    1.     27.    V600,000.     29.    ef  in. 

Article  148 

1.  ^  +  n:r;  2  /ITT-  5  ,  (2  n  +  1)  TT  +  I  ;  ±  5  ±  2  utt;  2  nx.    3.  2  titt  -  41°  48', 
4  D  5         3 

(2  n  +  1)  TT  +  41°  48';  (2  71  +  1)  TT  ±  70°  32';  63°  26'  +  mr;  2  nir  +  11°  32'; 

(2  n  +  1)  TT  -  11°  32'.     5.    68°  12'  +  mr;  2mr  -  16°  35';  (2  n  +  1)  tt  +  16° 

35';2n7r  ±  5°  44'. 


290 


ANSWERS 
Article  150 


sin 

cos 

tan 

CSC 

sec 

.  cot 

1. 

-h 

±  W3 

^i 

-2 

-I. 

1 

V3 

3. 

±! 

±1 

1 

±f 

±f 

1 

5. 

±1 

^f 

1 
V3 

±2 

^1 

V3 

7. 

±i\ 

-f£ 

±j% 

±v- 

-H 

±%' 

9. 

.6 

n 
m 

h 

±1 

±1 
1          "" 

1 
m 
ji 
1 
h 

±1 
1          ^ 

±1    ' 

m 

y,„2  _  ffi 

n 

11. 

V»i2  -  n2 
,          1 

1<l 

+  Vl  -  /l2 

Vl  -  /i2 

h 

16. 

"^02 +62 

2ab 
a2  +  62 

^     2ab 

^  a2  +  &2 
^  a2  -  62 

a2+62 
2a6. 

2  ah 

Article  151 


<      ,         tan  X  _   2  Vcsc2x  —  1      _  „      ^ ,- r— 

1.    ±  -pr=  3.   ^^^ z 5.   COS  »  ±  Vl  -  C0S2  0. 

Vl  +  tan2  X  csc2  a; 


Article  159 


1.  1;  0.     3.  iM;  Ml.     5.   ± 


VS  ±  4  V2 


V5  T  4  V2 
9 


Ill 


[(12  ±  63)  ±  (18  ±  14)  V3].  9.  iffil.  11.  Jf.  13.  ±  |§i|,  ±  mi. 
±  nil;  IM-  15.  ^-  17.  i.  19.  sin  202r  =  J  V2  -  V2;  cos  202^  = 
-§V2  +  V2;  tan  202|°  =  V3  -  2  V2,  sinT^  =  l'^2\J2  -  \JS  -  1;  cos  7*° 
=  I  V2  V2  +  V3  +  1;     tan  7^  =    Vl5  +  8  VS  -  10  V2  -  6  VO. 


Vl5  +  8  V3  -  10  V2 

Article  160 


1.   4°  40',  3°  20'.    3.   8;  5. 


1.   2  nvr  ± 


Article  166 

3.  {n  +  l)7r.   5.  /iTT +45°;n7r  +  71°34'.   7.  2  nx  +  36°  52'. 


9.    nir.    11.    nir;  iiTT  ± -:.    13.    nw;nTr±-.     15.    (2  n  +  1) 

19.  -^^  ;  ^.  21.  n:r;  135°  +  nvr.  23.    2  nw 
r  —  s'  r  +  s 

+  30°;   (2n +  l)7r  -  30°.   27. 


17. 


4n± 


K^  "■  - 

-^^8-     ^••2(p±5)"- 

60°; 

2  MTT  -  120°.  25.   2  TiTT 

(2  n  +  1)  1  ±  30°. 

ANSWERS 

Article  168 

V2 
5 

1. 

r  = 

±5,6- 

=  tan-if. 

3.  r  =  ±  41,  0  = 

tan- 

-IV.    5.  r  = 

=  ± 

,11" 
,n- 

-1  1. 
-11. 

7.    r  =  ±  3  V5,  9 
11.  x2  +  2/2  =  r2. 

=  tan-i(-  3).   9 

13.  "1  +  i  =  1. 

a-        62 

.  r  = 
15. 

=  5  V2,  <^  = 
a;2  +  2/2  + 

tan 

1. 

291 


i,e 


Article  179 

1.  Area  =  4828,  A  =  97°  48',  B  =  18°  21'.  8,  C  =  63°  50'.  2.  3.  Area  = 
1445.7,  A  =  34°  24',  B  =  73°  15',  C  =  72°  21'.  6.  6  =  290.9,  c  =  289.0, 
B  =  72°  6'.  7.  6  =  5340,  c  =  6535,  A  =  81°  52'.  9.  a  =  9548,  c  =  10804, 
C  =  105°  59'.  11.  No  solution.  13.  c  =  3120,  c'  =  402.2,  B  =  26°  52',  B' 
=  153°  8'j  C  =  131°  47',  C  =  5°  31'.  15.  h  =  .5458,  h'  =  .1814,  A  =  39° 
37',  A'  =  140°  23',  5  =  117°  51',  B'  =  17°  5'.  17.  c  =  .7105,  A  =  76°  20',  B 
=  44°  53',  Area  =  .2024.  19.  a  =  13.534,  B  =  15°  9'.  4,  C  =  131°  19'.  6, 
Area  =  32. 564.  21.  A  =  149°  49',  B  =  3°  2',  C  =  27°  9'.  23.  B  =  51°  9',  B' 
=  128°  51',  C  =  87°  38',  C  =  9°  56',  c  =  116. 82,  c'  =  20. 172.  25.  b  =  71760, 
B  =  146°  43',  C  =  14°  4'.  27.  A  =  57°  53',  B  =  70°  17',  C=  51°  50'.  29.  c  = 
38088,  B  =  48°  34'.  7,  C  =  49°  38'. 3.  31.  A  =  18°  12',  B  =  135°  51',  C  =  25° 
57'.  33.  c  =  748. 1,  A  =  42°  51',  B  =  64°  9'.  35.  b  =  .000331,  B  =  83°  33', 
C  =  32°  36'.  37.  c  =  2406,  c'  =  227.6,  B  =  31°  58',  B'  =  148°  2'.  C  =  120° 
44',  C  =  4°  40'.  39.  c  =  369.27,  A  =  39°  39'.6,  C  =  90°.  63.  7;  Vl29; 
20V3.  65.6824.  67.  45°,  60°,  75°;  612.5  ft.;  683  ft.  69.698.3  ft.  71.121ft.; 
390  ft.     73.1145  ft.     75.8640  ft.     77.62.00  ft.     79.969.2  ft.     81.19955  m. 

83.  59.1;  513.  85.  25,  33^,  41§.  87.^^^.  89.  18. 76  chains;  7.578  acres. 
91.  3.620  acres,  south  of  dividing  line.  93.  10.802  chains  east  of  A.  95.  i  = 
tan-ii'g.     97.  20°  7'.     99.  12°  32'. 

Article  183 
1.  55;  403.     3.  14;  200.    5.  28;  364.    7.  p  -  V  9;  20p  -  95 g.    9.  I  =  150; 
d  =  3.     11.  a  =  9;  d  =  2.     13.  a  =  IS;  d  =  5.    15.  a  =  17;  Z  =  97.    17.  a  =  |; 
I  =  -%".     19.   n  =  16,  1  =  69.     21.  n  =  14;   a  =  12.     23.  n  =  103,  a  =  1281. 
25.  8925.     27.  10  sec.     29.  29700  ft. 

Article  187 

1.  I  =  256;  S  =  508.  3.  I  =  4096;  S  =  5461.  5.  I  =  -262I44  '  ^  =  ^62li4' 
7.  /  =  a  (1  +x)7;  S  =  "  ^  +  ^)^  "  «  .•  9.  ±  45;  288,  ±  1728.   11.  ±  12,  4, 

±  5,  g,  ±  2*7-  13.  12,  3,  f,  A-  15.  a  =  2,  S  =  254.  17.  a  =  6,  S  =  W. 
19.  n  =  6,  S  =  126.  21.  r  =  3,  n  =  7.  23.  r  =  I,  n  =  6.  25.  n  =  9, 
I  =  19683.  27.  a  =  5,  Z  =  320. 


292  ANSWERS 

Article  189 
1.  3f.     3.  \\     6.  f.     7.  8  sec. 

Article  191 

1.   o  =  115  or  1;  d  =  -10  or  +2.       3.  c  =  -11   or  ^P;  d  =  4  or  -  ^'' . 
5.  First  number  V;  com.  diff.  ^\  V2989  or  r\  V  -  1779.     7.  Middle  num- 


y  H5  62)±y/9  64 


ber  =  6;  com.  diff.  =  ±y  H5  62)±i/9  64 +  -^.    9.55°,  60°,    65°.      11.   a, 

ar^,  ar,  ar^,  ....      13.  i  i,    ±  10,  ±  40,  ±  160.      15.  10.11  inches.      17. 
$1845X1010.    19.  2  a;  4  a  Vs. 

Article  194 

A 
1.  $2975  +  .      3.  $1489  +  .      5.  20.     7.  $497.8C 


r  (1  +r)'"-i 
Article  203 

1.  Convergent.     3.  Conv.  if  |  x  i  <  1.     Div.  if  1  .x  |  ^  1.     5.  Conv.  if  |  x  |<  ^  • 

7.  Conv.  if  1  <  X  <  10.     9.  Convergent.     11.  Convergent  for  all  values  of  x. 
13.  Conv.  for  all  values  of  x.     15.  Conv.  when  —  1  <  x  S  1. 

Article  205 
1.  .41.     3.  1.261.     5.  .0589  +  .     7.  .0053  +  . 


1.  tx2.     3.3x2-1.     5.  ± ^=r-    7.  ±— ^^=-    9.  ±    j-J^- 

"  2V^^  Vx2-1  s/x^-l 


Article 

209 

3 

'-v.-fc 

2V^x 

Article 

214 

9x' 

_  J 1_ 

2  x^      3  x^ 

1.12x3  +  15x2.     3.  — ^  +  ~'^-    5.   --4--^,-    7.2x6=='-.     9.   -sinx 
4x*      9x'  2x^      3x^ 

6x 
+  sec  X  tan  x.     11.    -   ^  _     •    13.   —  sin  x  tan  x  +  cos  x  log  cos  x.     15.  sec2  x. 

17.  sec  X  CSC  X. 

Article  216 

1.  3  x2  +  2  X.     3.  cos2  X  —  sin2  x.     5?  c^.    7.  12  I,  where  I  =  length  of  edge. 
Area    of    base.      11.   s 13.    6/ 

Z  IT 

6  —  c  COS  A    da      c  —  b  cos  A     da  _bc  sin  A 


).    Area    of    base.      11.   ?^I^.      13.    6^2  +  3;   603;   9;   3.      15.    | 


dc  a  dA 


ANSWERS  293 

Article  222 

a  +  ^-^+  ■■■''-  l+^-|-|+      ■■■'''  1  +  ^-^^^  + 
J^  x3  +  •  •  •  .      15.    1  -  1  X  +  f  a:2  -  ^  +  •  •  •  .     17.  1  +  2  X  +  2  a;2  + 

2x3+..  •.     19.  3^--^_-i^,-l^„+...   .     21.2-^4--^, 
2-3^      4.3'^'      8-3'^  7-2?      49-2^ 

_^0x^^+....       23.    lrV2-7^+6V3^  +  -JL--i^_+---1. 
343-2'''  6^L  V3x       V27x3  J 

1  4x^       ,     14x4  i40a;2 

25.   1 f  H 5 r%  +  •  •  •   . 

(16a)*       (16a)"       (16a)«       (16  a)'^ 

Article  229 

1.  i  n3  +  ^  n2  +  1  n.     3.  ^  n3  +  I  n2  +  i  n.    5.    nia  +  ^?-^  d    \  • 

Article  231 

3.    .0314;  .0204.    5.   S'' 16'"  OS'-'.oO;  18"  48' 10".  1. 

Article  232 

1.    Sixth  entry  should  be  .364.     3.    Sixth  entry  should  be  3' 30". 

Article  234 

1.  an  =  an-2-  an-V,  n>l.     3.  1  +  x2  +  x3  +  2  x4  +  •  •  •  .     5.  i  X  - 

2  X        4  ^    +  2  ^    -t-  •     '•  3  X        9    ^  27  81      ^ 

Article  239 
.  3  _  1  3     2 2_        _1_       g        2  V3+3 

^■8(3x  +  l)       8(x  +  3)'      'x       x+2'^x-2'      •6(x-2-V3) 

_        2  V3  -  3  -    _JL 1 1__      9    3       5-3x 

6  (X  _  2  +  V3)  4  (X  -  1)       4  (X  +  1)       2  (x2  +  1)  ■  ^-  X  "^  x2  +  4* 

n.  1-1    +.,AF^..   13.   -2+2      _        3,  ^g_  ^ 


3  X  ^    3  (x2  +  3)  X    '    X  -  2        (x  -  2)2      *"'  3  (x  +  1) 

2  17...    1    ,.+.^^  +  ..^^T^-    19.      4  1 


3(x2-x  +  l)  2(x-l)    '   5(x-2)    '    10(x  +  3)  x-2       x-1 

21    _1 2_        ^_, 

X  +  1    •   X  +  2  ^  X  -  2 


294  ANSWERS 

Article  241 

3.  X  =  11,  y  =  I,  2  =  t\.     5.  X  =  H,  2/  =  1.  2  =  ft-     7.  Not  independent. 

Article  249 

1.  0.  3.  0.  5.  8.  7.  398.  9.  832.  11.  aihcsdi.  33.  u  =  j%%,  v  =  -  i, 
10  =  If.     35.  Inconsistent. 

Article  257 

1.  V2,  -  45°;  5,  36°  52';  ^146,  114°  27';  2,  90°;  2,  0°;  2,  0°;  6,  30°;  36, 
-60°;  4,90°. 

Article  259 

3.  ±  3;  ±  3  i.  5.  xi  =  2;  X2  =  2  (cos  72°  +  i  sin  72°)|  X3  =  2  (cos  144° 
+  isin  144°);  etc.     7.  xi  =  V3;  X2  =  -^^-^= — ;  X3  =  — ^^"2 ;  etc. 

Article  260 

1.  3  cos2  dBmO  -  sin3  0.  3.  cos*  ^  —  6  cos2  0  sin2  d  +  sin*  0.  5.  6  cos^  0  sin  6 
-  20 cos3 0 sin3 6  +  Qcose sin^ 0. 

Article  263 
1.  24.     3.  240. 

Article  264 
1.  20.     3.  120.     7.  190. 

Article  266 

1.  1260.  3.  360.  5.  eC*  X  14C7  +  eCs  X  nCe  +  eCs  X  14C5  =  71500.  7.  73. 
9.  4;  there  will  be  three  different  throws.  11.  36;  there  will  be  21  different 
throws. 

Article  268 

1.  j%.  .  3.  j\.    6.  i§.    7.  ^h-    9-  .'/s-    11.  i^lhz- 

Article  270 
1.  A.    3.  /j;  i    5.  6.    7.  x*5.    9-  i- 

Article  275 

1.  a;3  -  6  x2  +  11  .T  -  6  =  0.  3.  x4  -  2  x3  -  4  x2  +  8  X  =  0.  6.  6  x*  - 
5x3-5x2+5x  -1=0. 

Article  280 
1.  -  1,  2,  2.     3.  3,  3,-2,  -  2.     5.  3,  3,  -  1,  -  2.     7.  1,  1,  1,  -  2. 
9.  3,  3,  ±  i 


ANSWERS  29^ 

Article  286 

1.  x3  +  2  a;2  -  4  X  -  8  =  0.  3.  x3  -  12  x  -  14  =  0.  5.  x2  +  2  x  +  1  =  0. 
7.  j;>  _  2  X  -  2  =  0.  9.  x3  +  x2  -  9  =  0.  11.  x3  -  9  x2  +  24  x  -  16  =  0. 
13.  x*  +  6  x3  +  x2  -  24  X  +  16  =  0.  15.  /i  =  1;  x3  -  3  x  =  0.  17.  /i  =  1 ; 
a;3  _  9x  -  7  =  0.     23.  ±  V^,  2.     25.  2,   ±  2  V2,  -  1  ±  V^- 

Article  291 
1.2,2,-1.    3.  1, -§,  -|.    5.  3,2  ±2  Vs.    7.-1,-2,3.    9.1,-1, 
-  1  ±  V^^.      llj  (-  1  ±  VB),  H5  ±  V37).      13.  2,  -  2,   -  2,   -  2. 

15.  4,  2,  -1  ±  V-3. 

Article  305 
1.  A   =  79°  aO'.S,  B  =  46°  15'.3,  C  =  70°  55'.6.   3.  A  =  130°  5'.4, 

5  =  32°  26'.1,  C  =  36°  45'.8.  5.  o  =  96°  24'.5,  6  =  68°  27'.4,  c  =  87°  31'.6. 
7.  c  =  50°  6',  A  =  129°  58',  B  =  34°  30'.  9.  a  =  43°  18',  B  =  28°  48', 
C  =  74°  22'.  11.  b  =  78°  17',  c  =  126°  46',  A  =  96°  46'.  13.  a  =  76°  25', 

6  =  58°  19',  C  =  116° 31'.  15.  a  =  124°  12'31",  c=  97°  12' 25",  B=  51°  18' 11". 
17.  a  =  58°  8'  19",  B  =  98°  20'  0",  C  =  63°  40'  0.  19.  b  =  75°  29',  c  =  108° 
14',  C  =  46°  52'.  21.  No  solution.  23.  c  =  84°  30',  B  =  56°  20',  C  =  97°  19'. 
25.  B  =  42°  37'  18",  137°  22'  42",  C  =  160°  1'  24",  50°  18'  55",  c  =  153° 
38'  42";  90°  5'  41".  27.  a  =  64°  23'  20",  b  =  99°  48'  50",  A  =  65°  33'  10". 

Article  306 

1.  N.Y.  -  S.F.  2568  mi.  N.Y.  -  M.C.  2090  mi.  S.F.-M.C.  1889  mi. 
Angles:  N.Y.  48°  58',  S.F.  55°  48',  M.C.  82°  40'.  Area:  2025300  sq.  mi. 
3.  X  =  g''  34'«  15«,  ^=  22°  6'  N;  course,  S  44°  28'  VV. 

Article  307 

1.  A  =  ±  92°  50';  <  =  ±  5^  4^^  12«.  3.  /i  =  43°  27';  A  =  70°  3'.  5.  N  1° 
33'.6  E  or  W;  <  =  ±  S''  55'"  54*.    7.  9°  46'.4. 


INDEX 


PAOB 

Abscissa 42 

Addition 4 

Altitude 281 

Alternating  series 173 

Annuities 169 

Antecedent 88 

Approximations 199 

to  the  roots  of  an  equation. .  260 

Area  of  plane  A 148 

of  spherical  A 277 

Arithmetic  progression 161 

mean 162 

Azimuth 281 

Base  of  logarithms 189 

Binomial  series 196 

convergence 196-7 

Binomial  theorem 33,  195 

Celestial  poles 280 

sphere 280 

equator 280 

Chance 244 

Circle 67 

Circular  measure 113 

Circular  parts,  Napier's  rules  of      270 

Co-factor 223 

Combinations 242 

Complex  numbers 21,  233 

Comparison  test 174 

Complementary  function.  . .     97,  100 

Computation 199 

of  logarithms 201-2 

Conic  sections 72 

Conjugate  complex  numbers.  . .       21 

Consequent 88 

Convergence  of  series 171 

of  binomial  series 196-7 

297 


PAGB 

Coordinates 42 

polar 231 

Cosecant 95,  100 

Cosine 95,  100 

Cotangent 95,  100 

Co  versed  sine Ill 

Cubic  equation 264 

Declination 281 

Degree  of  a  term 64 

of  a  polynomial 64 

De  Moivrc's  theorem 235 

Derivatives 184 

higher 192 

formulas 186 

Determinants,  general  definition  220 

of  second  order 217 

of  third  order 218 

properties 222 

use  in  sodving  equations 226 

Differences 203 

Difference  quotient 180 

Discriminant  of  quadratic  oquii- 

tion 57 

of  cubic  equation 266 

Division 5 

synthetic 255 

Ellipse 68 

Equations,  cubic 264 

exponential 86 

linear 37-53 

of  nth  degree 249-63 

quadratic 54-86 

quartic 267 

trigonometric 137 

Equator,  celestial 280 

Evolution 18 


298 


INDEX 


PAGE 

Exponent,  irrational 20 

laws 17-21 

negative 17 

positive  integral 7 

rational 19 

zero 18 

Exponential  equations 86 

values  of  sin  a;  and  cos  x  . . . .  239 

Extremes 88 

Factor,  highest  common 11 

theorem 10,  249 

Factoring 9 

Fractions 13 

partial 213 

Functions 90,  179 

continuous 180 

hyperbolic 240 

trigonometric 103 

inverse  trigonometric 134 

Geometric  mean 164 

progression 163 

infinite  progression 165 

series 173 

Graphic  solution  of  linear  equa- 
tions       39-50 

of  quadratic  equations.  .      65-80 

Graph  of  straight  line 41,  43 

of  trigonometric  functions.  .  .      105 

Harmonic  progression 167 

mean 167 

Highest  common  factor 11 

Horizon 281 

Hour  angle 281 

Hyperbola 70 

rectangular 71 

Hyperbolic  functions 240 

Imaginary  number 21 

Infinite  series 171 

solution  of  linear  equations.         38 

Infinity 6 

Initial  line 99,  231 

Integral  expression 14 


PAGE 

Interest 168 

Interpolation 206 

Inverse  ratio 88 

trigonometric  functions 134 

variation 91 

Involution 17 

Irrational  expression 20 

exponent 20 

number 19 

Joint  variation 92 

Law  of  sines 144 

of  cosines 145 

of  tangents 146 

Least  common  multiple 11 

Limit 171 

Linear  equations 37 

graphic  solution 39-50 

simultaneous 46 

Logarithms 28,  30 

computation  of 201-2 

laws  of 30 

modulus  of 203 

natural  or  Naperian ■.  .  .  189 

Maclaurin's  series 193 

Mean  arithmetic 162 

geometric 164 

harmonic 167 

proportional 89 

Means,  in  a  proportion 88 

Meridian 281 

Modulus  of  common  logarithms  203 

Multiplication 4 

Naperian  logarithms 189 

Napier's  rules  of  circular  parts  270 

analogies 275 

Natural  logarithms 189 

Numbers,  complex 21 

conjugate  comple:: 21 

imaginary 21 

irrational 19 

principal  root  of 22 

rational 5 


INDEX 


299 


PAGE 

Numbers,  real 20 

surd 20 

Ordinate 42 

Parabola 59,69 

Partial  fractions 213 

Permutations 242 

Polar  coordinates 231 

triangle 269 

Pole 231 

Power 8 

Present  worth 169 

Principal    value    of   an   inverse 

trigonometric  function .  .  135 

of  a  root 22 

Progressions,  arithmetic 161 

geometric 163 

infinite  geometric 165 

harmonic 167 

Proportion 88 

Quadratic  equations 54-86 

formula 56 

simultaneous 64 

Quartic  equation 267 

Radian 143 

measure 143 

Radius  vector 231 

Ratio 88 

inverse 88 

Rational  expression 5 

exponent 19 

number 5 

Real  number 20 

Root  of  an  equation 56 

principal 22 

RooLs  of  unity 237 

Secant 95,  100 

Series,  alternating 173 

binomial 196 

geometric 173 

infinite 171 


PAGE 

Series,  Maclaurin's 193 

power 173 

ratio  test 176 

Sine 95,  100 

Slope 181 

Sphere,  celestial 280 

terrestrial 279 

Spherical  excess 269 

triangles 269-79 

Straight  line ,. 41,  43 

Subtraction 4 

Surd  expression 20 

number 20 

Sjmthetic  division 255 

Tangent,  trigonometric .  .  .       95,  100 

to  a  curve 181 

Terminal  line 99 

Terrestrial  sphere 279 

Triangles,  plane  right 98 

plane  obUquc 144-155 

spherical  right 270-1 

spherical  oblique 272-8 

Trigonometric  equations 197 

Trigonometric  functions  . .  .      94-140 

defined 95,  100 

discontinuities 104 

graphs 105 

inverse 134 

line  values 101 

periodicity 106 

signs 101 

variation 103 

Undetermined  coefficients 211 

Variable 90 

Variation 90 

direct 91 

joint 92 

inverse 91 

Versed  sine Ill 

Zero 5 

exponent IS 


APPENDIX   A 

The  Greek  Alphabet 

Letters. 

Name. 

Letters. 

Name. 

Letters. 

Name. 

A,  a, 

Alpha 

1,1, 

Iota 

P,P, 

Rho 

B,/3, 

Beta 

K,  ., 

Kappa 

S,a, 

Sigma 

r,7, 

Gamma 

A,  X, 

Lambda 

T,  T, 

Tau 

A,  5, 

Delta 

M,  M, 

Mu 

Y,v, 

Upsilon 

E,., 

Epsilon 

N,  ., 

Nu 

*I^  </>, 

Phi 

z,r, 

Zeta 

H,  ^, 

Xi 

X,  X, 

Chi 

H,  r/, 

Eta 

0,0, 

Omicron 

^.'A, 

Psi 

e,  d,  ■&, 

Theta 

n,7r, 

Pi 

n,  CO, 

Omega 

List  of  Formulas 

Factors  of  a"  ±  b",  n  being  a  positive  integer  (9). 
a"  —  6"  is  divisible  by  (a  —  6)  and  by  (a  -\-  b)  when  n  is  even, 
a"  —  6"  is  divisible  by  (a  —  6),  not  by  (a  -\-b),  when  w  is  odd. 
o"  +  &"  is  divisible  by  (a  +  6),  not  by  (a  —  b),  when  w  is  odd. 
a"  +  6"  is  not  divisible  by  (a  +  b)  or  by  (a  —  b)  when  n  is  even. 


Special  Cases. 

a2  _  52  =  (a  +  &)  (a 


6) .     a^  +  6^  has  no  real  factors. 
a3-b^=(a-b)(a^-\-ab-i-b^). 
a3  +  63  =  (a  +  b)  (a^  -  a6  +  62). 


64=(a2  +  62)  (a2-62). 


a. 


+  6"*  has  no  real  factors. 


i5  _  55  =  (a  _  I)  (^4  _|_  ^36  +  a-b~  +  a63  +  6-*). 


o5  +  65  =  (a  +  6)  (a4 


+  o262-a63+64). 


Factor  Theorem.  —  If  /  (x)  reduces  to  zero  when  x  =  a,  f  (x) 
contains  the  factor  (x  —  a).     (11),  (272). 

301 


302  FORMULAS 

Exponents.    (20)  to  (25). 


1  i      xn 


(a-)y  =  a'y.      {ahY  =  a'h\     if)  =f,' 
Imaginary  or  Complex  Numbers.     (26.) 

t^V^;      i2=-l;      1-3  =-i;      2;4=+l,   etc. 

V^  =  i  yfa.      a^  +  h^  ={a  +  ib)  (a  -  ib). 
X  +  iy  =  r  (cos  d  -\- i  sin  6)  =  re^^. 

Surds.  —  If  a  -\-  ^  =  c-{-  Vd,   where  V6  and  V5  are  surds, 
then  a  =  c  and  b  =  d.     (29.) 
Logarithms.     (37),  (39),  (226). 
If  a^  =  m,  then  x  =  loga  rn. 

171 

loga  mn  =  logo  m  +  loga  n.  loga  ^  =  loga  w  -  logo  n. 

loga  mP  =  p  loga  m.  loga  V  W  =  -  loga  W, 

loga  a  =  1.     loga  1=0.  k)ga  0  =  -  »,  if  a  >  1. 


Change  of  Base,      loga  w  =  logb  m  X  loga  &• 
If       a  =  10  and  b  =  e,  then  loga  b  =  login  e  =  M.     (Table  V.) 
Hence  logio  m  =  M  loge  m. 

Binomial  Theorem.     (42),  (220-1). 

/    ,  7x„       „  ,       «   11.  ,  w(w— 1)  „_o72  I  n(n— l)(n— 2)  „_„  „ 
(a+6)"  =  a"+na"-i6H —  ,^    ' a'^' -b^ -{- -^ r~ -'a"  ^b^+- 

n(n-l)(n-2)  .  .  ■  (n-r+D  ^,,_,^,  j 

\r 


FORMULAS  303 

Quadratic  Equation,  a.c-  -\-bx-\-c  =  0.     (74),  (76),  (78). 

Roots  real  and  unequal  if  6^  —  4  ac  >  0. 


Ellipse:  5;  +  ^  =  1.       Hyperbola:  ^  -  ^  =  ±  1. 


:=  -h±^b^-4ac^       Roots  real  and  equal  if      b^  -  4  ac  =  0. 

^  Roots  imaginary  if  6^  —  4  ac  <  0. 

b  c 

Sum  of  roots  = .     Product  of  roots  =  -  • 

a  a 

Graph  of  ?/  =  ax^  +  6a:  +  c  is  a  parabola. 
Standard  Equations  of  Conic  Sections. 

Circle:     x'^  i- y^  =  r^-     Parabola:      y^  =  4ax;    rc^  =  4  ay. 

^-r^  =  L       Hyperbola:  ^;  -  |. 
Rectangular  Hyperbola:  xy=±k^. 
Ratio,  Proportion,  Variation. 
If,  a:b  =  c:d, 

then, 

(1)  a  +  b:b  =  c  +  d:d; 

(2)  a-b:b  =  c-d:d; 

(3)  a-{-b:  a  —  b  =  c  -]-  d:  c  —  d; 

(4)  a"  :  6"  =  c'^ :  d\ 

If  ai :  bi  =  a2  :  bo  =  as  :  bs  =   •  •  ■  , 

,,  .  ^,  , .  pai  +  ga2  +  ras  +  ■ 

then  anv  of  these  ratios  =  -r — ; — r — , — r — , — 

pbi  +  962  +  ?-63  +  ■ 

where  p,  q,  r  are  any  multipliers; 


1  r  iu  ^-  ,7^1"  +  ap"  +  «3"  +    ■    •    • 

also  any  of  these  ratios  =  V/  ,  „  ,   ,  „    ,  ,  ^   1 

T  Oi    -f-  02    -r  03    -+-••• 

If  y  <x  X    then     y  =  kx; 

If  ?/  «  -     then    y  =  -,     or    xy  =  k. 

Arithmetic  Progression.     (180.) 

a  =  first  term;       d  =  common  diff.;       n  =  number  of  terms; 
I  =  last  or  nth  term;       S  =  sum  of  ?i  terms, 
nth  term  =  Z  =  a  +  (n  —  1)  c?. 

S  =  ^(a-^l)=n(a  +  ''~-dy 
Arithmetic  mean  of  a  and  b  =  — ?i — 


304  FORMULAS 

Geometric  Progression.     (184.) 

r  =  the  ratio;   a,  n,  I,  S,  as  above. 

nth  term  =  I  =  ar'^'''-. 

„         1  —  r"      a  —  rl 

o  =  a- = 

1  —  r         1  —  r 

Geometric  mean  of  a  and  6  =  Va6. 
Sum  of  infinite  geom.  progr.  =     _    ,  if  \r\<  1. 

Infinite  Series.  —  Tests  for  convergence  or  divergence. 

,    Series,       mi  +  W2  +  ws  +  •  •  •  +Un-i  +  Un+  •  •  •     . 

Converges   when   the   terms   are   alternately   +   and   — ,  and 

steadily  decrease  toward  zero  (199). 

u 
Converges  when  the  ratio  — —  becomes  and  remains  numeri- 

Un-l 

cally  less  than   1   for  all   values  of   n,   provided   always   that 
lim  Un  =  0.     (202.) 

u 
i     Diverges  when  the  ratio  — ^  becomes  and  remains  greater  than 

Un-l 

1,  or  approaches  1  from  the  upper  side.     (202.) 

Converges  when  its  terms  are  numerically  less  than  the  corre- 
sponding terms  of  a  series  known  to  converge  absolutely.     (201.) 

Diverges  when  its  terms  are  all  of  like  sign  and  are  numerically 
greater  than  the  corresponding  terms  of  a  known  divergent  series. 


Test  Series. 

1  -{- X  +  X"  -\- x^  -\- 


\  conv.  when  ]  x  ]  <  1 ; 
I  div.     when  |  x  |  =  1. 

1        1        1  ( conv.  when  p>l; 

P  +  2P      3^       '  '  '     (div.     whenp=l. 

Derivatives.     (210.) 

_dij  _  y      Ay  ^  =  slope  of  tangent  to  curve  y  =f(x). 
^^  ^  dx^  Ax™o  A^   (  =  rate  of  change  of  y  relative  to  x. 


FORMULAS  305 

Formulas  for  Differentiation.     (211-2.) 

du  _  du    dy  ^_n  ^  ^^y^  —    ^^ 

dx      dy    dx  dx        '  dx  dx 

d  (u  -\-  V  -\-  w  -\-  ■   •  ■)  _  du      dv       dw 
dx  dx      dx      dx 

,  /u\        du  _     dv^ 
d  (uv)  _     dv         du  \v/  _     dx  dx 

dx  dx        dx         dx  v- 

dy  _dy    du 
dx      du   dx 

when  y  is  a  function  of  u,  and  u  a  function  of  x. 


dx                          dx 

1       da'        , , 

-  •     -T-  =  a   log  a. 

X       dx              ^ 

dsin  X 

— y —  =  cos  X. 
dx 

d  COS  X 

— 3 =  —  sin  X. 

dx 

dtanx           „ 

— J =  sec^  X. 

dx 

d  cot  X               „ 

dsec  X               ,          , 
— -. —  =  sec  X  tan  x  dx. 
dx 

d  cscx 

— J —  =  —  esc  a;  cot  x. 

dsm-^x            1 

d  cos"  ^  X           —  1 

dx            Vl  -  x^ 

dx            Vl  -  x^ 

d tan" ^  X          1 

dcot-^x         -1 

dx           l+a;2 

dx           l-\-x^ 

dspc-irc             1 

dcsG-'^x           -1 

dx           a:  Va:2  -  1 

dx           x  Vx2  -  1 

Maclaurin's  Series.     (218.) 

/  Cr)  =  /  (0)  +  xf  (0)  +  -j^  /"(O)  +  ^^/'"  (0)  +  .  .  .     . 

Some  Standard  Series. 

e'  =  1  -j-  X  -\-  -.^ -{-  j^z -{-  ■  •  •    .     Always  convergent. 

sina:  =  x  —  jir  +  -jT—   ••'    .  Always  convergent. 

\6       [5 


306 


FORMULAS 


loge  (l  +  x)  =  x-'j-\-j- 


Always  convergent. 

Convergent  only  if 
-  l<a;=  1. 


Theorem  of  Undetermined  Coefficients.     (233-4.) 

If,  for  all  values  of  x  from  x  =  Otox  =  h  where  h  is  any  number 
other  than  zero,  we  have 

ao  +  aix  +  a2X^  +  •  •  •   +  ^n^"  +  .  •  .    =0, 

then  flo  =  0,    ai  =  0,    a2  =  0  •  •  •  an  =  0,  •  •  •     . 

If,  for  values  of  x  as  above,  we  have 

ao  +  dix  +  a2X^  +  •  •  •  =ho  +  bix  +  b2X^  +  .  .  .    , 

then  ao  =  bo,    ai  =  6i,    a-z  =  62,  etc. 

Partial  Fractions.  (235-8.)  —  The  partial  fractions  may  be 
determined  according  to  the  factors  of  the  denominator  of  the 
given  fraction. by  the  following  rules: 

Corresponding  fraction  or  fractions: 

A 


Form  of  factor: 
{ax  +  b), 


ax  -j-  b 


{ax  +  6)", 
{ax^  -\-  bx  -^c), 


Ai 


^^ L     . 

ax  +  b  '   {ax  +  b)'^^ 
A.  +  5 


I 


{ax^-{-bx-\-c) 


^  Aix  +  B, 


ax^  -\-bx  -\-  c 
A2X  +  B2      , 


'ax^+bx-\-c  '  (ax2+6a;+c)2 
Determinants.     (240-9.) 

ai62  ~  a2&i. 


{ax  +  by 


ArnX  +  Brn 

{ax^-\-bx-\-c)^ 


ai  61 

a2  62 

ax  61  ci 

a2  62  C2 

as 

&3    C3I 

=  a\Ai  —  biBx  +  c\Ci 

=  a\b2Cz  +  azb'sCi  +  a36iC2 

—  036201  —  0261 C3  —  aibsC2. 


FORMULAS 


307 


=  ai  Ai  —  biBi  +  ciCi  —  diDi, 


Here  A\,  Bi,  C\,  are  the  minors  of  oi,  h\,  ci,  respectively, 
ai  61  c\  d\ 
a2  62  Co  do 
as  &3  C3  ds 

04    64    C4    C?4 

where  ^1,  5i,  C\,  Di  are  the  minors  of  ai,  6i,  ci,  c?i,  respectively. 
Similarly  for  a  determinant  of  any  order. 

Differences  and  Interpolation.     (227-32.) 

Let  uo,  III,  U2,  ■  ■  ■  he  a,  given  sequence,  and  let  AiWo,  A2i^o, 
A3U0,  •  •  •  be  the  first  terms  of  the  successive  difference  columns. 
Also  let  „Ci,  „C2,  Tic's,  •  •  •  be  the  binomial  coefficients,  i.e., 


,Ci 


^C2 


n  (n  —  1) 


iCs  = 


nin-1)  (n-2) 


etc. 


|2  "^"^  |3 

Let  Un  be  the  nth  term  of  the  sequence  and  s„  the  sum  of  its 
first  n  terms.     Then 

Un  =  U0+  nCiAiWo  +  nC2^2U0  +  nC^AsUQ  +    '     '     '     ', 
Sn  •=  nClUo  +  nCoAiWo  +  nC^AoUo  +  „C4A4Wo  +    •     •     •    . 

If  Uo  =  f  (xo),  ui  =  / (xo  -{-h),U2=f(xo-\-2h),U3  =  f  (.To -\-Sh), 
.  .  .  ,  then 

f(Xo  +  nh)=f{Xo)-\-nCiAif(Xo)-\-nC2A2f(Xo)-\-nC3A3f(Xo)+   •    '    "   • 

Here  71  need  not  be  an  integer. 
Useful  Approximations.     (224.) 
When  X,  y,u,v,  .  .  .  are  small  (near  0)  we  have,  approximately, 


(l+a:)(l+2/)  =  l-{-x  +  tj. 

(l+x)(l-y)  =  l+x-  y. 

{\  -  x)  {\  -  y)  =  1  -  X  -  y. 

1  +  2/ 


1 
1  +x 

1 


=  \-x. 


l+x. 


l+x-y. 


{l+u){\+v) 
(1  +  a:)"  =  1  +  nx. 
Vl+x  =\  +  hx. 

'        =1       ' 


Vl+x  2 

(l+x)2  =  l+2x. 


1  -a; 

=  l+a:+?/+  •  •  •  -u—v- 

As  special  cases  of  this: 
Vl-a;  =  l-ix. 


==  =  1  +  -a:. 
-a:  2 

a:)-  =  1  -2x. 


vr 
(1- 


e'  =  l+x.     log,  (1  +  x)  =  a:,     logio  (1  +  a;)  =  .43 a;, 
sin  X  =  tana;  =  a;  (radians).  cosx  =  1. 


308 

FORMULAS 

More  accurately, 

smx  =  x--- 

-v2 

COS  a:  =  1  -  y 

tana:  =  re  +  — • 

o 

De  Moivre's  Theorem.     (256.) 

(cos  0  +  i  sin  0)"  =  COS  nO  +  i  sin  n9. 

2»  =  r"  (cos  n0  +  i  sin  n^). 

The  »ith  Roots  of  Unity.      (259.) 

a;^;  =  cos h  ^  sin  ,        k  =  0,  I,  2,   .  .  .  ,  n  —  1. 

Expansions  of  cos  n6   and  sin  nQ.     (260.) 

cos  nd  =  cos"  6 ^-r^ — -  cos"~2  Q  sin2  d 

,  n  (n  -  1)  (n  -  2)  (w  -  3)       „  ^  ^   .  ^ 
+  ^ 1^     ^  ^ ^  cos'^-^  ^  sm^^  -  .  . 

sin  nd  =  n  cos""  ^  ^  sin  ^  -  ^  (^  ~  1)  (^  "  ^)  ^os'^-s  ^  gin^  6 -\-  ■   ■ 

\^ 
Exponential  Values  of  sin  x  and  cos  x.     (261.) 

sm  a;  = ^r-. cos  a;  = 

2i  2 


Hyperbolic  Functions.     (262.) 


3         ^5 


.  ,  e"  —  e  -^  ,   a;-^   ,  a;     , 

sinha:  =  — 2— =  a:+j3  +  j^+.  • 

■    C"^  ~\~  6~ '''  X  X 

cosha:  =  — 2— =l+j2  +  |4  +  -   • 

,     ,  sinh  X  ^,  cosh  x 

tanh  X  =  — s coth  x  =  ^—, — 

co^  X  sinh  X 

sech  X  =  — i csch  x  =  ^^ 


cosh  a:  sinh  x 

Permutations  and  Combinations.     (263-4.) 

nPr  =  n  (n  -  1)  (n  -  2)  .  .  .  (n-r+1).     „P„  =  |_n. 
^        „P.      n  (n  -  1)  .  .  .   (n  -  r  +  1)  _ 

\L  \l 


Plane  Trigonometry 

Definitions.     (124,  132.)  —  In  right  triangle  ABC,  whose  sides 

are  a,  h,  c  [figure  of  (124)], 

•  A  C-  A  b  J.  A  (^ 

Sin  A  =  -  >      cos  A  =  -}      tan  A  =  r' 
ceo 

A  C  ^  C  .     A  ^ 

esc  A  =  -  >       sec  A  =  r  >       cot  A  =  -  • 
aba 

vers  A.  =  1  —  cos  A.     covers  A  =  1  —  sin  A. 

]\Iore  generally,  if  x  be  an  angle  of  any  magnitude,  as  XOP  in 
the  figure  of  (132), 

ordinate  abscissa       ^  ordinate 

sm  X  =  -7T-, 1     cos  X  =  -p— >      tan  x  =  —, — ; — ■  > 

distance  distance  abscissa 

distance  distance  ^         abscissa 

cscx  =  — p — — )      seca;  =  -i — ; >      cota;  =  — p — -— • 

ordinate  abscissa  ordinate 

Relations    between   the   Functions   of   an   Angle.     Formulas, 
Group  A.    (137.) 

1       •  1  o    4-  1  e        *         COS  a: 

1.  sinx  = 3.  tanx  =  — ^-  5.    cot  a:  =  ^ 

CSC  X  cot  X  Sin  X 

1  sin.r  6.   sin- .r  +  cos-a;  =  1. 

2.  cosa;=- 4.   tana:  = '- ■         «     .   ,  ^     o  ., 

sec  X  cos  X  7.1  +  tan-  x  =  sec-  x. 

8.1+  cot^  X  =  CSC-  X. 

Rules  for  expressing  any  function  of  any  angle  in  terms  of  a 
function  of  an  acute  angle..    (139.) 

Any   function    of    any   angle  x  is  numerically   equal    to    the 

( same  function     .      .  ,        ,...,,,  (even       ,,. 

<       .       ,.  of  a:  increased  or  diminished  by  any  {     ,  ,   multi- 

(  co-function  *  (  odd 

pie  of  90°. 

The  sign  of  the  result  must  be  determined  according  to  the 

quadrant  of  x. 


310  FORMULAS 

Functions  of  +  aj  and  -  x.     (140.) 

/(+  x)  =/(—  x),  when/  =  cosine  or  secant." 
/(+x)  =  —  S {—  x),  when/  =  sine,  cosecant,  tangent,  cotangent. 
Angles  Corresponding  to  a  Given  Function.     (146.) 
Let  6  denote  the  smallest  positive  angle  having  a  given  func- 
tion equal  to  a  given  number  a.     Then  all  angles  such  that 

^    (  sin  x  =  a  ,     , 

I.  ]  are    X  =  2  WTT  +  ^    and     (2  n  +  1)  tt  -  ^; 


■■■) 


esc  a:  =  a 
cos  x  =  a 
sec  X  =  a 


are    x  =  2mT  ±  6] 


^_._.      tan  X  =  a 

111,  <  are    x  =  nir  -{-  6. 

(cot  X  =  a 

Formulas,  Group  B.     (155.) 
9.  sin  {x  -\-  y)  =  sin  x  cosy  +  cos  x  sin  y. 

10.  cos  (x  +  y)  =  cos  X  cosy  —  sin  x  sin  y. 

11.  sin  {x  —  y)  =  sin  a:  cos  ?/  —  cos  a:  sin  y. 

12.  cos  (x  —  y)  =  cos  a;  cos  ?/  +  sin  a;  sin  ?/. 

,        tanx  +  tany 

13.  t'"'(^  +  !/)  =  i-tanxtani/- 

14.  cot(x  +  y)^"°V"°*^7'- 

^        ^  cot  a;  +  cot  ?/ 

tan  a;  -  tan  ?/ 

15.  t^^(^-^)  =  l+tanxtant/- 

cot  X  cot  y  +  1 

16.  cot(x-^)=    ,3ty-cotx 

Formulas,  Group  C.     (157.) 

Double  Angle.  Half-Angle. 

14.   sin  2  X  =  2  sin  x  cos  x.  17.   sin 


.=±v/^- 


15.  cos  2x  =  cos^x  -  sin2  x,        18.    cos  ^  x  =  ±  v/^  +cosx^ 

=  l-2sin2x,  j9_   tan|x=±v/iZ^, 

T   1  +  cos  X 
=  2cos2x-l.  _^l-cosx^ 

sin  X 

16.  tan2x  =  ^-^*J54_.  =       ^^"  ^     ! 

1  —  tan^  X  1  —  cos  X 


FORMULAS  311 

Fonnulas,  Group  D.     (158.) 

20.  sin  w  +  sin  y  =  2  sin  — - —  cos  — ^ — 

21.  smw  —  sm  y  =  2cos — ^ — sin  — ^r — 

22.  cos  w  +  cosy  =  2  cos — ^ — cos — - — 

oo  ^     .     U-\-  V    .     U  —  V 

23.  cosw  —  cosy  =  —  2  sin — - — sin  — ^ — 

Solution  of  Plane  Triangles 

Right  Triangles.  —  By  means  of  the  definitions  of  the  trigo- 
nometric functions  write  an  equation  involving  the  two  given 
parts  and  a  required  part;  solve  this  for  the  required  part. 

Oblique  Plane  Triangles.     (169-172.) 

Law  of  Sines:  1.   a  :  6  :  c  =  sin  ^  :  sin  B  :  sin  C  (169) 

Law  of  Cosines:      2.   a^  =  62 .+  c^  -2  be  cos  A.  (170) 

Law  of  Tangents:    3.   ^  =  ^"f^^Tm-  (171) 

"^  a-\-b      tan  ^  {A  -\-  B) 

Half-Angles.     (172.) 


Let    s  =  |(a  +  6+c)     and    ,  =  y/(^  -  «)  (^  -  ^)  (^  "  ^) . 


4.   sin  5  A 


'  oc  ▼        s  (s  —  a) 

^^5^.  7.   tan*4=-^- 

▼        6c  "  s  —  a 


Solution  of  Oblique  Plane  Triangles.     (173-8.) 

Case  L   Given  two  angles  and  a  side.  (174) 

Use  law  of  sines. 

Case  II.   Given  two  sides  and  the  included  angle.  (175) 

Use  law  of  tangents,  then  law  of  sines. 


312  FORMULAS 

Case  III.    Given  two  sides  and  an  opposite  angle.  (176) 

Use  law  of  sines.     Ambiguous  case. 

Case  IV.   Given  the  three  sides.  (177) 

Use  one  of  the  formulas  (4),  (5),  (6),  or  (7)  above, 
preferably  the  last  one. 


Area  =  |  a6  sin  C  =  -^sis-a)  (s-h){s-c).         (178) 


Spherical  Trigonometry 

Spherical  Right  Triangle.    (313-6.)  —  Let  A,  B,C  be  the  angles, 

and  a,  h,  c  the  sides.     Arrange  the  five  parts  a,  b,  co-B,  co-c,  co-A 

in  circular  order.     These  parts  are  then  connected  by  Napier's 

Rules : 

,     .  ,  ,,  (  product  of  cosines  of  opposite  parts  ; 

sme  of  middle  part  =  j  ^^^^^^^  ^^  ^^^^^^^^^  ^^  ^^.^^^^^  p^^^^ 

To  solve  a  spherical  right  triangle  use  Napier's  Rules  to  write 
a  formula  involving  the  two  given  parts  and  a  required  part. 
To  solve  a  quadrantal  triangle,  solve  its  polar  right  triangle. 

Spherical  Oblique  Triangles.     (317-22.) 

Law  of  Sines :  sin  a  :  sin  &  :  sin  c  =  sin  A  :  sin  B  :  sin  C. 
Law  of  Cosines:  cos  a  =  cos  6  cos  c  +  sin  6  sin  c  cos  A. 

Half-Angles. 


-(a  +  6  +  c);  tanr 


.  /sin  (s  —  a)  sin  (s  —  b)  sin  (s  —  c)  ^ 
V  sin  ,s 


4. 


,    1   .        .  /sm  (s  —  b)  sm  (s  —  c) 
2  T  sm  6  sm  c 


1  .       .  /sm  s  sm  (s  —  a) 

5.  cos  7;  A.  =  V -. — j-\ ' 

2  V        sm  b  sm  c 

6.  tan  p:  A 


1  /sin  (s  —  b)  sin  (s  —  c)  ^ 

2  ~  V       sin  s  sin  (s  —  a) 


1  .  tan  r 

tan^A  = 


2  sin  (s  —  a) 


I 


FORMULAS  313 

Half-Sides. 


S  =  l(A+B  +  C);  tan  72=  y  ^^^  ^ 


2'      '       '  ^'  V  cos  (S  -  A)  cos  (S-B)  cos  (S-C) 

13. 


.    1         J-cosScos(S-  A) 
sin  ;^  a  =  y  - 


2         V  sin  B  sin  C 

14.  cos  o  a  =  V  •    D    •    ^ 

2  >  sin  5  sin  C 


1         .  /cos  (.S  -  B)  cos  (>S  -  C) 
sin  5  sinC 

/—  cos  S  cos  (*S  —  ^ ) 
cos  {S  -  B)  cos  {S-C) 


,  _  ,1  4  /     —  COS  »S  COS  (*S  —  ^ ) 

15.  tan^ra  =  V  ~ 


16.  tan  -rt  =  tan  R  cos  {S  —  A). 

Napier's  Analogies. 

in  +      1  ^        j,A        sin H^  -  -6)  ,       1 

19  tan  -  (a  -  6)    =    .    {)  .    ,      'z  tan  -  c. 

2  sin  I  (vl  +  i?)        2 

or>  J.  1  /        ,     r\  cos  I  (^—5),  1 

20.  ta„2(«+6)    =,„4;^^g;tan^o. 

21.  tani(A-B)=$4f^cotic. 

2  sin  I  (a  +  6)        2 

22.  tan  -{A  +  B)= f^. — —~  cot  ^  C. 

2  '  cos  ^  (a  +  &)        2 

Spherical  Excess. 

E  =(A-j-B-^C)-  180°. 
1  „  tan  I  a  tan  §  6  sin  C 


23.      tan^S  = 


2  1  +  tan  I  a  tan  ^  &  cos  C 

24.      tan  2  -2^  =  Vtan  |  s  tan  !(«  —  «)  tan  |(s  -  6)  tan^  (s  -  c). 

Area  =  ^  ^^^Iq^^^''  X  4  7ri?^  =  7^;  (radians)  X  R'. 

Solution  of  Spherical  Oblique  Triangle.     (323.) 

I.   Given  two  sides  and  an  opposite  angle. 

Use  law  of  sines,  then  Napier's  Analogies.     Two  solu- 
tions possible. 
II.   Given  two  angles  and  an  opposite  side. 
As  in  I. 


314  FORMULAS 

III.  Given  the  three  sides. 

Use  formulas  for  the  half-angles. 

IV.  Given  the  three  angles. 

Use  formulas  for  the  half-sides. 

V.  Given  two  sides  and  their  included  angle. 

Use  Napier's  Analogies,  then  law  of  sines. 

VI.  Given  two  angles  and  their  included  side. 

As  in  V. 


APPENDIX  B 

Explanation  of  the  Tables  and  Their  Use 
TABLE  I 

This  table  gives  the  decimal  part,  or  mantissa,  of  the  logarithm 
of  every  positive  number  containing  not  more  than  three  sig- 
nificant figures.  The  mantissas  of  the  logarithms  of  numbers 
containing  more  than  three  significant  figures  are  to  be  obtained 
by  interpolation  (35).  The  integral  part,  or  characteristic,  of  the 
logarithm  must  be  supplied  by  the  computer,  according  to  the 
position  of  the  decimal  point  in  the  number. 

Rules  for  Characteristics. 

(a)  When  the  number  has  n  significant  figures  to  the  left  of 
the  decimal  point,  the  characteristic  of  its  logarithm  is  n  —  1. 

(b)  When  the  number  is  a  decimal  with  n  ciphers  between  the 
decimal  point  and  the  first  digit  which  is  not  zero,  the  characteris- 
tic of  its  logarithm  is  9  —  7i,  and  — 10  must  be  supplied  to  com- 
plete the  logarithm. 

The  reason  for  these  rules  will  become  evident  when  we  consider 
an  example. 

Example.  Let  us  find  log  302.  In  the  table  find  30  in  the 
left-hand  column  and  run  across  the  page  horizontally  to  the 
column  headed  2.     There  we  find  that 

mantissa  of  log  302  =  .4800. 

Now  302  lies  between  100  and  1000,  i.e.  between  10^  and  10^. 
Hence,  by  the  definition  of  a  logarithm,  log  302  must  lie  between 
2  and  3.     Therefore  the  characteristic  is  2,  and 

log  302  =  2.4800. 

This  is  of  course  not  the  exact  logarithm  of  302,  but  only  its  value 
to  four  decimal  places. 

Writing  the  kst  equation  in  exponential  form,  we  have 

302  =  los-isoo^ 
315 


316  EXPLANATION   OF    TABLES 

Multiplying  both  sides  by  10, 

3020  =  10  X  102-4800  =  103.4800,     Hence,  log  3020=  3.4800. 

Multiplying  again  by  10, 

30200  =  10  X  103-4800  =  i04-48oo_     Hence,  log  30200  =  4.4800. 

Therefore,  -where  a  number  is  multiplied  by  10,  the  character- 
istic of  its  logarithm  is  increased  by  1;  the  mantissa  remains 
unchanged. 

Dividing  the  above  equation  successively  by  10,  we  obtain 
30.2  =  102-4800      ^  10  =  101-4800^ 
3.02  =  10^-4800      4-  10  =  100-4800, 
.302  =  100-4800       ^  10  =  100-4800-1^ 

.0302  =    100-1800-1  -^  10  =   100-4800-2^ 
.00302  =   100-4800-2  ^  10  =   100-4800-3^ 

and  so  on.     As  logarithmic  equations  these  are: 

log  30.2  =  1.4800, 
log  3.02  =  0.4800, 

log  .302  =  0.4800  -  1  =  9.4800  -  10, 

log  .0302  =  0.4800  -  2  =  8.4800  -  10, 

log  .00302  =  0.4800  -  3  =  7.4800  -  10, 

and  so  on.     The  second  form  in  the  last  three  equations  is  used 
for  convenience  in  computations;  it  is  in  accordance  with  rule  (b). 
To  discuss  rules  (a)  and  (b)  more  generally,  let  m  be  any  number. 
Then  by  the  definition  of  a  logarithm,  when 

m  lies  between  log  m  lies  between 

(1)  1  and  10,  0  and  1, 

(2)  10  and  100,  1  and  2, 

(3)  100  and  1000,  2  and  3, 

(4)  1000  and  10000,  3  and  4, 

and  so  on.     Therefore,  when  m  has 

(1)  1  digit  to  the  left  of  the  point,    log  m  =  0.+  •  • 

(2)  2  digits  to  the  left  of  the  point,  log  m  =  l.-\-  •  • 

(3)  3  digits  to  the  left  of  the  point,  log  w  =  2.+  •  • 

(4)  4  digits  to  the  left  of  the  point,  log  m  =  3.+  •  • 

and  so  on..   Hence  rule  (a). 


EXPLANATION   OF   TABLES  317 

In  the  case  of  decimal  numbers, 

when  m  Hos  between  log  m  lies  between 

(1)  1  and  0.1,  0  and  -  1, 

(2)  O.I  and  0.01,  -  I  and  -  2, 
.  (3)              0.01  and  0.001,  -  2  and  -  3, 

(4)  0.001  and  0.0001,  -  3  and  -  4, 

and  so  on.     That  is,  when  m  is  a  decimal  number  in  which 

(1)  no  cipher  follows  the  point,  log  m  =  9.+  •   •  •  —  10 

(2)  1     cipher  follows  the  point,  log  ?/i  =  8.+  ■  •  •  —  10 

(3)  2     ciphers  follow  the  point,  log  /«  =  ?.+  •   •  •  —  10 

(4)  3     ciphers  follow  the  point,  log //i  =  6.+  •   •  •  —  10 
and  so  on.     Hence  rule  (b). 

Interpolation.  —  Exam-pie.     Find  log  3024. 
From  the  table, 

mantissa  of  log  302  =.4800;    ^^^^^^^^^  ^    0014 
mantissa  of  log  303  =  .4814; 

Assuming  that  the  increase  in  the  logarithm  is  proportional 
to  the  increase  in  the  number,  we  have 

mantissa  of  log  3024  =.4800  +.4  X.0014  =.4806. 

The  result  is  here  given  to  the  nearest  unit  in  the  fourth  decimal 
place,  .4  X.0014  being  taken  equal  to  .0006  in  place  of  .00056. 

Proportional  Parts.  —  For  convenience  in  interpolation,  the 
tabular  differences  greater  than  20  are  subdivided  into  tenths  and 
tabulated  under  the  heading  "  Prop.  Parts."  When  the  difference 
is  less  than  20,  the  interpolation  is  best  made  mentally.  If  it  is 
desired,  the  table  of  proportional  parts  may  be  used  when  d  <  20 
by  taking  half  the  proportional  part  corresponding  to  double  the 
difference. 

Examples. 

1.  log  164.3  =  ? 

Mantissa  oflog  164  =    .2148;     d  =  27, 

Correction  for  .3  =  S 

log  164.3  =  2.2156 

2.  log  (164.3) '  =  ? 

log  (164.3)^  =  I  log  164.3. 

=  I  (2.2156)  =  1.4771. 


318  EXPLANATION  OF  TABLES 

3.        log  .01047  =  ? 

Mantissa  of  log  104  =     .0170;        d  =  i2, 
Correction  for  .7  =         29 

log  .01047  =  8.0199  -  10 


log  V(.01047)4  =  ?_ 


-V^.01Q47^  =  (.01047)^ 
log  \/(.01047)^  =  I  log  (.01047)^ 

=  I  (8.0199-10). 
4  (8.0199  -  10)  =  32.0796  -  40  =*  22.0796  -  30. 
^  (22.0796  -  30)  =  7.3599  -  10. 

Note.  When  a  logarithm  which  is  followed  by  —10  is  to  be  divided  by  a 
number,  add  and  subtract  a  multiple  of  ten  so  that  the  quotient  will  come 
out  in  a  form  followed  by  —10.     Thus: 

i  (8.2448  -  10)  =  1  (38.2448  -  40)  =  9.5612-  10. 

Anti-logarithm.  —  The  number  whose  logarithm  is  x  is  called 
the  anti-log aritlnn  of  x. 

Thus,  a  X  =  log  7n,  then  m  =  anti-log  x. 

Given  a  logarithm,  to  obtain  the  corresponding  number  {anti-loga- 
rithm) . 

Examples. 

1.  log  m  =  0.4806.     m  =  ? 

The  given  logarithm  lies  between  the  tabular  logarithms  .4800  and  .4814, 
to  which  correspond  the  numbers  302  and  303  respectively.     Thus  we  have 
Number.  Mantissa  of  log. 

302  .4800  I     ^ 

m  .4806  i     [  14    ' 

303  .4814        ) 

Hence,  without  regard  to  the  decimal  point,     m  =  302  +  fj  =  3024 +  . 
Pointing  o£f  properly, 

TO  =  anti-log  0.4806  =  3.024+. 

2.  log  TO  =  7.0959  -  10.       TO  =  ? 
mantissa  of  log  124  =  .0934  j        ^ 
mantissa  of  log    to  =  .0959  \       [35 
mantissa  of  log  125  =  .0969  > 

Hence  m  has  the  sequence  of  figures 

124  +  U  =  1247  +. 
Pointing  off  properly, 

TO  =  anti-log  (7.0959  -  10)  =.0012474-. 

Note.  The  value  of  the  quotient  1 1  may  be  obtained  from  the  column  of 
Prop.  Parts  by  finding  the  number  of  tenths  of  35  required  to  equal  25.  We 
have  from  this  column, 

.7  X  35  =  24.5  and  .8  X  35  =  28.0. 


EXPLANATION  OF  TABLES  319 

Hence  we  see  that  to  make  25  we  need  a  little  more  than  .  7  X  35.     A  close 
approximation  would  be  .71+,  making  m  =.0012471+. 

When  the  tabular  (lifference  is  largo,  it  is  possible  to  obtain  correctly  more 
than  four  significant  figures  of  a  number  when  its  four-place  logarithm  is  given. 

Cologarithm.  —  The  cologaritkm  of  a  number  is  the  logarithm 
of  the  reciprocal  of  the  number. 

Thus:  colog  m  =  log—  =  log  1  —  log  wi  =  —  log  m. 

In  practice  we  usually  write  it  in  the  form 

colog  771  =  —  log  m  =  (10  —  log  7n)  —  10. 

Rule.     To    form   the   cologarithm  of  a  number,  kubtract  its 
logarithm  from  10  and  write  — 10  after  the  result. 

Exmnples. 

1.  colog  302  =  (10  -  log  302)  -  10 

=  (10  -  2.4S00)  -  10  =  7..5200  -  10. 

2.  colog  .003024  =  (10  -  log  .003024)  -  10 

=  (10  -  [7.4806  -  10])  -  10  =  2. 5194. 

Use  of  the  Cologarithm. 

Exa7nple.     Calculate  the  value  of  • 

541  X  •  0o2o 

Let  m  be  the  value  of  the  given  fraction.     Then  without  the  use 

of  cologarithms  the  calculation  is  as  follows. 

log  m  =  log  302  +  log  .415  -  log  541  -  log  .0828. 
log  302  =    2.4800  ■      log  541  =    2.7332 

log  .415  =    9.6180  -  10  log  .0828  =    8.9180  -  10 

12.0980  -  10  11.6512  -  10 

11.6512-  10 
log  m  =    0.4468,  m  =  2.7975. 

To  use  cologarithms,  we  write 

m  =  302x.415X5|jX^3- 

log  m  =  log  302  +  log.  415  +-  colog  541  +  colog  .0828 
log  302=    2.4800 
log  .415=    9.6180-10 
colog  541  =    7.2668  -  10 
colog  .0828  =    1.0820 

log  771  =  20.4468  -  20 
m  =    2.7975. 


320  EXPLANATION  OF  TABLES 

As  a  last  example,  we  calculate  the  value  of  the  quantity, 


=n/: 


(■00812)i  X  (-47L2)3 
(-522.3)3  X  (.01242)? 


[ 

To  take  account  of  the  signs,'  which  must  be  done  independ- 
ently of  the  logarithmic  calculation,  we  note  that  the  cube  of  a 
negative  quantity  occurs  on  both  sides  of  the  fraction;   hence  the 
sign  of  the  fraction  is  plus. 
We  now  write 

logm  =  I  [log  (.00812)3  -|_  log  (471.2)3  +  colog  (522.3)3 
+  colog(.01242)U 
log  .00812  =  7.9096  -  10  log  (.00812)^  =  8.6064  -  10 

log  471.2  =  2.6732  log  (471.2)3     =8.0196 

log  522.3  =  2.7179  log  (522.3)3     =  8.1537 

log  .01242  =  8.0941  -  10  log  (.01242)*  =  8.5706  -  10 


Hence 


log  (.00812)5  = 

8.6064  - 

-10 

log  (471.2)3    = 

8.0196 

colog  (522.3)3     = 

1.8463  - 

-  10 

colog  (.01242)^  = 

1.4294 

2 

|19.9017  - 

-20 

log  m  = 

9.9508  - 

-10 

m  = 

.8929. 

Exercises.     Verify  the  following  equations: 


1.  log  7  =  0.8451. 

2.  log  253  =  2.4031. 

3.  log  253.5  =  2.4040. 

4.  log  .0253  =  8.4031  -  10. 

5.  log  .002533  =  7.4036  -  10 

6.  log  6544  =  3.8158. 

7.  log  4.007  =  0.6028. 

8.  log  .9995  =  9.9998  -  10. 

9.  log  V766  =  1.4421. 


10.  log  yig  =  7.1158  -  10. 

11.  log  (.0022)3  =  2.0272  -  10. 

12.  log  <JW22  =  9.1141  -  10. 

13.  log  (.01401)"  =  8.5171  -  10. 

14.  log  (.0003684)1  =  7.9S20  -  20. 

15.  colog  200  =  7.6990  -  10. 

16.  colog  .7  =  0.1549. 

17.  colog  0448^  1.3487. 

18.  colog  V5475  =  8.1308  -  10. 


\.  ^   Xr- 


26. 

</ 

-  .0822 

=  -  .4348. 

27. 

(- 

6.213)' 

=  2.076. 

28. 

( 

^Ir— 

29. 

Jn 

1 

.32)1 

.05761. 

EXPLANATION  OF  TABLES  321 

19.  colog  (.0003684)=  =  12.0180. 

20.  antilog  1.2222  =  10.68. 

21.  antilog  3.0675  =  4650. 

22.  antilog  0.4000  =  2.5118. 

23.  antilog  (8.3250  -  10)  =    .021135. 

24.  antilog  (6.9525  -  10)  =  .0008964. 

25.  (.748)3  =  .4185. 

TABLE  IL 

This  table  gives  the  logarithms  of  the  sine,  cosine,  tangent  and 
cotangent  of  angles  from  0°  to  90°,  at  intervals  of  10', 

^yhen  the  angle  is  taken  from  the  left-hand  column  of  the  page, 
the  name  of  the  function  must  be  sought  at  the  top  of  the  page; 
when  the  angle  is  taken  from  the  right-hand  column  of  the  page,  the 
name  of  the  function  must  be  sought  at  the  foot  of  the  page. 

When  the  function  is  numerically  less  than  1,  —10  must  be 
written  after  its  tabular  logarithm.  This  is  the  case  with  the 
sines  and  cosines  of  all  angles  between  0°  and  90°,  with  tangents 
of  angles  between  0°  and  45°,  and  with  cotangents  between  45° 
and  90°. 

For  convenience  in  interpolation  the  differences  of  the  tabular 
logarithms  are  given,  and  these  differences  are  subdivided  into 
tenths  in  the  column  of  proportional  parts.  Hence  this  column 
contains  the  corrections  to  the  tabular  logarithms  for  each  minute 
of  angle  from  1'  to  9'  inclusive.  These  corrections  are  to  be 
added  when  the  logarithm  increases  with  the  angle,  and  they 
are  to  be  subtracted  when  the  logarithm  decreases  as  the  angle 
increases. 

When  the  logarithm  of  a  function  of  an  angle  greater  than  90** 
is  required,  change  to  the  equivalent  function  of  an  angle  less  than 
90°  (139).  Algebraic  signs  must  be  adjusted  independently  of 
the  logarithmic  calculation,  as  in  the  use  of  Table  I. 

Seconds  of  arc  must  be  reduced  to  the  equivalent  fractions  of  a 
minute  of  arc. 

To  obtain  log  sec  x,  take  from  the  table  colog  cos  x;  for  log 
CSC  X  use  colog  sin  x. 
Examples. 
1.         log  sin  20°  13'  =  ? 

log  sin  20°  10'  =  9. 5375;        d  =  34. 
d  for  3'  (Prop.  Parts)  =  10.2 

log  sin  20°  13'  =9.5385  -  10. 


322  EXPLANATION  OF  TABLES 

2.  log  cos  20°  13'  =  ? 

log  cos  20°  10'  =  9. 0725;        d  =  4. 
d  for  3'  =  4  X  .3  =  1.2 

log  cos  20°  13'  =  9.9724  -  10. 

3.  log  tan  29°  47'  =  ? 

log  tan  29°  40'  =9. 7556;        d  =  29. 
d  for  7'  (Prop.  Parts)   =  20.3 

log  tan  29°  47'  =  9.7576  -  10 

The  same  result  may  also  be  obtained  by  starting  with  log  tan  29°  50',  thus: 
log  tan  29°  50'  =9.  7585;         d  =  29. 

d  for  3'  =  8.7 

log  tan  29°  47'  =  9. 7576  -  10. 

As  a  rule,  in  interpolating  start  from  the  nearest  tabular  number. 

4.  log  cot  29°  47'  =  ? 

29. 


log  cot  29°  50' 

=  0.2415;         d 

d  for  3' 

8.7 

log  cot  29°  47' 

=  0.2424. 

log  sin  58°  44'  =  ? 

log  sin  58°  40' 

=  9.9315;        d 

d  for  4' 

3.2 

log  sin  58°  44' 

=  9.9318  -  10. 

log  tan  67°  23'.5  =  ? 

log  tan  67°  20' 

=  0.3792;         d 

d  for  3'.  5  =  10.8  +  1.8 

12.6 

log  tan  67°  23'. 5 

=  0.3805. 

Here  we  obtain  d  for  3'.5  from  d  for  3'  +  d  for  0'.5.      Note   that  d  for 
0.5  is  simply  one-tenth  of  d  for  5'. 

7.  log  cos  105°  51'.6  =  ? 

cos  105°  51'.6  =  -  sin  15°  51'.6. 

Neglecting  the  algebraic  sign  we  have 

log  sin  15°  50'  =  9.4359;         d  =  44. 

d  for  1'.6  =  7\0 

log  sin  15°  51'.6  =  9.4366  -  10  =  log  cos  105°  51'.6. 

8.  log  tan  250°  34' .3  =  ? 

tan  250°  34'. 3  =  tan  70°  34' .3. 
log  tan  70°  30'  =  0.4509;         d  =  40. 

d  for  4'.3  =  17.2 

log  tan  70°  34'.3  =  0.4526  =  log  tan  250°  34' .3. 


EXPLANATION    OF    TABLES  323 

Angles  near  0^  or  near  90°. 

When  an  angle,  x,  lies  near  0°,  sin  x,  tan  x,  and  cot  x  vary  too 
rapidly  with  x  to  permit  of  accurate  interpolation  of  their  loga- 
rithms from  the  table.  The  same  is  true  of  cos  x,  tan  x,  and  cot  z, 
when  x  lies  near  90°.  We  will  show  how  accurate  values  of  these 
logarithms  may  be  obtained. 

T   X                   „      ,      sin  a;         1    rp      1      tan  x 
Let  &  =  log and   T  =  log » 

X  being  expressed  in  minutes  of  arc. 
Then  log  sin  x  =  log  x'  +  S, 

and  log  tan  x  =  log  x'  -{- T. 

When  X  is  small  the  quantities  S  and  T  vary  quite  slowly  with  x. 
The  values  of  S  and  T  are  given  in  the  last  column  of  the  first 
page  of  Table  II,  x  ranging  from  0°  to  5°;  —10  is  to  be  added  to 
the  tabular  numbers  there  given. 

To  get  log  sin  x,  reduce  x  to  minutes  of  arc  and  take  log  x'  from 
Table  I;  to  this  logarithm  add  S. 

To  get  log  tan  x,  add  T  to  log  x'. 

To  get  log  cot  X,  first  get  log  tan  x  and  form  the  cologarithm  of 
the  result. 

For,  log  cot  X  =  colog  tan  x. 

To  obtain  log  cos  x,  log  tan  x  or  log  cot  x,  when  x  lies  between 
85°  and  90°,  calculate  the  co-function  of  the  complementary  angle 
by  the  method  given  above. 

To  find  the  angle  from  log  sin  x,  log  tan  x  or  log  cot  x,  when  x 
lies  near  0°,  we  use  the  relations 

log  x'  =  log  sin  X  —  S; 
log  x'  =  log  tan  X  —  T; 
log  x'  =  —  log  cot  X  —  T. 

The  necessary  values  of  .S  and  T  can  be  obtained  after  finding 
an  approximate  value  of  .r  from  Table  II. 

To  find  X  from  log  cos  x,  log  tan  x,  or  log  cot  x,  when  x  lies  near 
90°,  replace 

log  cos  x  by  log  sin  (90°  —  x) ; 
log  tan  X  by  log  cot  (90°  —  x) ; 
log  cot  X     by    log  tan  (90°  —  x). 


324  EXPLANATION    OF    TABLES 

Then  90°  —  x  can  be  obtained  by  the  method  given  above  for 
angles  near  0°.     Hence  x  is  determined. 

Examples. 

1.  Find  log  sin  x,  log  tan  x  and  log  cot  x  when  re  =  1°  22'  12". 

X  =  1°22'  12"  =  82'.2.  \ogx'  =  log  82.2  =  1.9149. 

logx  =  1.9149  log  a;   =  1.9149 

S  =  6. 4637  -  10  T  =  6.  4638  -  10 

log  sin  X  =  8. 3786  -  10  (    ■  log  tan  x  =  8. 3787  -  10  0    ^ 
log  cot  X  =  colog  tan  x  =  1.6213. 

2.  Find  log  cos  x,  log  tan  x  and  log  cot  x  when  x  =  89°  5'  50". 
Let  ?/  =  90°  -  X  =  54'  10"  =  54'.17. 

Then  log  cos  x,  log  tan  x,  log  cot  x  are  equal  respectively  to  log  sin  y,  log  cot  y, 
log  tan  y,  which  may  be  found  as  in  example  1. 

3.  log  sin  X  =  8.2142;      x  =  ? 

From  Table  II,   x  =  50'  +  ;         hence     S  =  6.4637  -  10. 
log  sin  X  =  8.2142  -  10 
S  =  6.4637  -  10 


logx' 

=  1.7505; 

X 

=  56 

.30 

=  56'  18". 

4. 

log  tan  X 

=  8.0804  - 

10;      X 

=  ? 

From 

Table   II,  X 

=  40'+  ; 

hence 

T 

=  6.4638 

log  tan  X 

=  8. 0804  - 

10 

T 

=  6.4638  - 

10 

1 

logx' 

=  1.6166; 

X 

=  41 

.36 

=  41'  21".6. 

6. 

log  cot  X 

=  8.6276  - 

10;       X 

= 

? 

H)'!^"-' 

Let 

V 

=  90°  -  X. 

'  --^fiO^r"^ 

Then 

log  tan  y 

=  log  cot  X 

=  8.6276 

— 

10. 

From  Table  II,  y 

=  2°  20'  + 

hence 

T 

=  6.4640. 

log  tan  y 

=  8.6276  - 

10 

T 

=  6.4640  - 

-  10 

logy' 

=  2.1636; 

/  =  145'. 

n 

=  2° 

25' 

44".        ,      if 

'^M 

Hence 

X 

=  90°  -  2/ 

=  87°  34 

16' 

'. 

VA  }y\ 

Let  the  student  obtain   the  results   required  in  the   last   five 
examples  by  direct  interpolation  from  Table  IL 
Exercises.     Veri'"y  the  following  equations : 


1. 

log  sin  20°  40' 

=  9.5477  - 

10. 

10. 

log  cos  81°  29' 

=  9.1706  - 

-10. 

2. 

log  cos  66°  30' 

=  9.6007  - 

10. 

11. 

log  cos  81°  31' 

=  9.1689  - 

-10 

3. 

log  tan  29°  35' 

=  9.7541  - 

-  10. 

12. 

log  cot  9°  6' 

=  0.7954. 

4. 

log  cot  37°  25' 

=  0.1163. 

13. 

log  sin  152°  27' 

=  9.6651  - 

-10 

5. 

log  sec  55°  50' 

=  0.2506. 

14. 

log  sin  2°  10'  10" 

=  8.5781  - 

-10 

6. 

log  esc  44°  50' 

=  0.1518. 

15. 

log  tan  1°  34'  20" 

=  8.4385  - 

-10 

7. 

log  tan  63°  27' 

=  0.3013. 

16. 

log  cot  0°  10' 22" 

=  2.5206. 

8. 

log  sin  81°  29' 

=  9.9952. 

17. 

log  cos  89°  28' 44 

'  =  7.9588- 

-10 

9. 

log  sin  81°  31' 

=  9.9952. 

^..^^ 

18. 

log  tan  88°  46' 14' 

'=  1.6683. 

EXPLANATION    OF    TABLi:S  325 

19.  log  sin  X  =  9.7926;  x  =  38°  20'. 

20.  log  sin  X  =  9.3548;  x  =  13°  5'. 

21.  log  sin  X  =  9.88G7;  x  =  50°  23'. 

22.  log  cos  X  =  9.G030;  x  =  66°  22'. 

23.  log  tanx  =  0.6278;  x  =  77°  44'.5. 

24.  log  cot  X  =  0.0906;  x  =  39°  4'. 

25.  log  cot  X  =  0.6648;  x  =  12°  12'.  5. 

26.  log  sec  X  =  0.1374;  x  =  43°  13'. 

27.  log  CSC  X  =  0.2890;  x  =  30°  56'. 

28.  log  sec  X  =  0.6680;  x  =  77°  35'.  8. 

29.  log  sin  X   =  8.3698;  x  =  1°  20' 34". 

30.  log  tan  x  =  8.7659;  x  =  3°  20'  18". 

31.  log  cot  X  =  1.2952;  x  =  2°  54' 3". 

32.  log  cos  X  =  8.5387;  x  =  88°  1'  8". 

33.  log  cot  X  =  7.9485;  x  =  89°  29'  28". 

34.  log  cscx  =  2.3549;  x  =  0°  15'  11". 

35.  log  sec  X  =  1.5102;  x  =  88°  13'  48". 

TABLE    III 

This  table  gives  the  numerical  values  of  the  six  trigonometric 
functions  of  angles  from  0°  to  90°  at  intervals  of  10'.  The  func- 
tions of  intermediate  angles  are  to  be  obtained  by  interpolation. 

By  using  the  tables  inversely,  an  angle  may  be  four-^^  u.aally 
to  the  nearest  minute,  when  a  function  of  the  angle  is  known  to 
four  decimal  places. 

TABLE   IV 

This  is  a  conversion  table  for  changing  from  sexagesimal  to 
radian  measure,  and  conversely.  The  entries  are  given  to  five 
decimal  places  in  radians,  corresponding  nearly  to  2"  in  sexagesi- 
mal measure. 

Examples. 

1.  Express  200°  44'  36"  in  radian  measure. 

200°  =  3  X  60°  +  20° 

3  X  60°  =  3  X  1.04720  =  3.14160  radians. 
20°  =  0.34907 

44'  =  0.01280 

36"  =  PX)0017_ 

200°  44'  36"  =  3.50364  radians. 

2.  Express  3.50364  radians  in  sexagesimal  measure. 


3.0 

radians  =  171°  53'  14" 

0.5 

"       =     28°  38' 52" 

0.003 

"       =           10'  19" 

0.0006 

"       =            2'     4" 

0.00004 

"      =                   8" 

3.50364  radians  =  200°  44'  37" 

326  EXPLANATION    OF   TABLES 

TABLE    V 

This  table  contains  the  values  of  a  number  of  mathematical 
constants,  generally  to  fifteen  places  of  decimals. 

TABLE    VI 

This  table  gives  the  values  of  the  natural  or  Naperian  loga- 
rithm of  X,  and  of  the  ascending  and  descending  exponential 
functions  e^  and  e""",  from  a;  =  0  to  a;  =  5  at  intervals  of  0.05. 
As  a  rule  the  tabular  entries  are  given  to  three  decimal  places. 

TABLE    Vn 

This  table  gives  the  values  of  n-,  n^,  Vn,  and  Vn,  for  values  of 
n  from  1  to  100. 

The  direct  use  of  the  table  requires  no  explanation.     As  an 

example  of  its  inverse  use  we  find  the  approximate  value  of  V320. 

We  have 

(6.8)3  =  314.432     (n  =  68), 

(6.9)3  ==  328.509     (n  =  69). 

Hence,  interpolating  linearly, 

(6.840)3  =  320  approx.,  or  V320  =  6.840+. 


TABLES 


328 


TABLE  I.     LOGARITHMS  OF  NUMBERS 


No. 
10 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Parts 

0000 

D043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

43 

42 

11 
12 
13 

0414 
0792 
1139 

3453 
D828 
1173 

0492 
0864 
1206 

0531 
0899 
1239 

0569 
0934 
1271 

0607 
0969 
1303 

0645 
1004 
1335 

0682 
1038 
1367 

0719 
1072 
1399 

0755 
1106 
1430 

2 
4 

8.6 
12.9 
17.2 
21  ,5 

8.4 
12.6 
16.8 
21.0 

14 
15 
16 

17 
1-8 
19 

1461 
1761 
2041 

2304 
2553 

2788 

1492 
1790 
2068 

2330 
2577 
2810 

1523 
1818 
2095 

2355 
2601 
2833 

1553 
1847 
2122 

2380 
2625 
2856 

1584 
1875 
2148 

2405 
2648 

2878 

1614 
1903 
2175 

2430 
2672 
2900 

1644 
1931 
2201 

2455 
2695 
2923 

1673 
1959 

2227 

2480 
2718 
2945 

1703 
1987 
2253 

2504 
2742 
2967 

1732 
2014 
2279 

2529 
2765 
2989 

6 

7 

9 
2 

25.8 
30.1 
34.4 
38.7 

25.2 
29.4 
33.6 
37.8 

41 

4.1 

8.2 

40 

4.0 
8.0 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

3 

4  , 

16:.4 

12.0 
16.0 

21 
22 
23 

3222 
3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 
3560 
3747 

3385 
3579 
3766 

3404 
3598 
3784 

6 
7 
8 
9 

24.6 
28.7 
32.8 
36.9 

24.0 
28.0 
32.0 
36.0 

24 
25 
26 

3802 
3979 
4150 

3820 
3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 

4082 
4249 

3927 
4099 
4265 

394b 
4116 
4281 

3962 
4133 
4298 

1 

3 
4 
5 
6 

39 

3.9 
7.8 
11.7 
15.6 
19.5 
23.4 

38 

3.8 
7.6 
11.4 
15.2 
19.0 
22.8 

27 
28 
29 

4314 
4472 
4624 

4330 
4487 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4378 
4533 
4683 

4393 
4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

4456 
4609 
4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

8 

31.2 

30^4 

31 
32 
33 

4914 
5051 
5185 

4928 
5065 
5198 

4942 
5079 
5211 

4955 
5092 
5224 

4969 
5105 
5237 

4983 
5119 
5250 

4997 
5132 
5263 

5011 
5145 
5276 

5024 
5159 
5289 

5038 
5172 
5302 

9 

35.1 

37 

3  7 

34.2 

36 

3.6 

34 
35 
36 

37 
38 
39 

5315 
5441 
5563 

5682 
5798 
5911 

5328 
5453 
5575 

5694 
5809 

5340 
5465 
5587 

5705 
5821 
5933 

5353 

5478 
5599 

5717 
5832 
5944 

5366 
5490 
5611 

5729 
5843 
5955 

5378 
5502 
5623 

5740 
5855 
5966 

5391 
5514 
5635 

5752 
5866 
5977 

5403 
5527 
5647 

5763 

5877 
5988 

5416 
5539 
5658 

5775 
5888 
5999 

5428 
5551 
5670 

5786 
5899 
6010 

2 
3 
4 
5 

6 
7 
8 
9 

7.4 
11.1 

14.8, 

18.5 

22.2 

25.9 

29.6 

33.3 

7.2 
10.8 
14.4 
18.0 
21.6 
25.2 
28.8 
32.4 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

35 

34 

41 
42 
43 

6128 
6232 
6335 

6138 
6243 
6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

1 
2 
3 
4 
5 

3.5 
7.0 
10.5 
14.0 
17  5 

3.4 
6.8 
10.2 
13.6 
17.0 

44 
45 
46 

47 
48 
49 

6435 
6532 
6628 

6721 
6812 
6902 

6444 
6542 
6637 

6730 
6821 
6911 

6454 
6551 
6646 

6739 
6830 
6920 

6464 
6561 
6656 

6749 
6839 
6928 

6474 
6571 
6665 

6758 
6848 
6937 

6484 
6580 
6675 

6767 
6857 
6946 

6.493 
6590 
6684 

6776 
6866 
6955 

6503 
6599 
6693 

6785 
6875 
6964 

6513 
6609 
6702 

6794 
6884 
6972 

6522 
6618 
6712 

6803 
6893 
6981 

7 
9 

1 

21.0 
24.5 
28.0 
31.5 

33 

3.3 
6.6 

20.4 
23.8 
27.2 
30.6 

32 

3.2 
6.4 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

3 
6 

9.9 
13.2 

9.6 

12.8 

51 

52 
53 

7076 
7160 
7243 

7084 
7168 
7251 

7093 
7177 
7259 

7101 
7185 
7267 

7110 
7193 

7275 

7118 
7202 

7284 

7126 
7210 
7292 

7135 
7218 
7300 

7143 
7226 
7308 

7152 
7235 
7316 

5 
6 
7 
8 
9 

16.5 
19.8 
23.1 
26.4 
29.7 

16.0 
19.2 
22.4 
25.6 
28.8 

54 
No. 

7324 

7332 

1 

7340 

7348 

7356 

7364 

7372 
6 

7380 

7388 

7396 

0 

2 

3 

4 

5 

7 

8 

9 

Prop.  Parts 

LOGARITHMS  OF  NUMBERS.     TABLE  I 


320 


No. 

55 

0 

7404 

1 
7412 

2 

7419 

3 

7427 

4 

7435 

6 

7443 

ti 

7 

8 

7466 

9 

7474 

Prop.  Parts 

7451 

7459 

31 

30 

56 

7482 

74^0 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

3  1 

3.0 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

6.2 
9.3 

6.0 
9.0 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

12.4 

12.0 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

15  5 

IS. (I 

15.0 
18.0 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

8 

:;i.7 
24.8 

21.0 
24.0 

61 
62 

7853 
7924 

7860 
7931 

7868 
7938 

7875 
7945 

7882 
7952 

7889 
7959 

7896 
7966 

7903 
7973 

7910 
7980 

7917 

7987 

9 

27.9 

27.0 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

29 

2.9 

5.8 

28 

2.8 
5.6 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

8.7 
11.6 
14.5 

8.4 
14'0 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

17.4 
20.3 

16.8 
19.6 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

8 

23.2 

22.4 

69 
70 

8388 
8451 

8395 

8457 

8401 
8463 

8407 
8470 

8414 
8476 

8420 

8482 

8426 

8488 

8432 
8494 

8439 
8500 

8445 
8506 

9 

26.1 

25.2 

27 

26 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

1 

2..  7 
5.4 

8.1 

2.6 
5.2 

7.8 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

3 

73 
74 

8633 
8692 

8639 
B698 

8645 
8704 

8651 
8710 

8657 
8716 

8663 

8722 

8669 

8727 

8675 
8733 

8681 
8739 

8686 
8745 

4 
5 

6 

10.8 
13.5 
16.2 

10.4 
13.0 

15,  C. 

75 

8751 

5756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

7 
8 
9 

18.9 
21.6 
24.3 

18.2 
20.8 
23.4 

76 

77 

8808 
8865 

B814 

S871 

8820 
8876 

8825 
8882 

8831 
8887 

8837 
8893 

8842 
8899 

8848 
8904 

8854 
8910 

8859 
8915 

78 

8921 

B927 

8032 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

25 

24 

79 

8976 

B982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

1 
2 

2.5 
5.0 

2.4 

4.8 

80 

9031 

)036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

3 
4 

7.5 
10.0 

9^6 

81 

9085 

)090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

5 

6 

7 

12.5 
15.0 
17.5 

12.0 
14  4 

82 

9138 

)143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

16:8 

S3 
84 

9191 
9243 

)196 

)248 

9201 
9253 

3206 
3258 

9212 
9263 

9217 
9269 

9222 
9274 

9227 
9279 

9232 
9284 

9238 
9289 

8 
9 

20.0 
22.5 

19.2 
21.6 

85 

9294 

)299 

9304 

3309 

3315 

9320 

9325 

9330 

9335 

9340 

23 

22 

86 

9345  < 

)350 

9355 

3360 

3365 

9370 

9375 

9380 

9385 

9390 

1 

2.3 

87 

9395  ' 

)400 

9405 

3410 

3415 

9420 

9425 

9430 

9435 

9440 

r. 

4.6 
6.9 

4.4 
6  6 

88 

9445 

M50 

9455 

3460 

9465 

9469 

9474 

9479 

9484 

9489 

9.2 

8.8 

89 

9494 

)499 

9594 

3509 

9513 

9518 

C523 

9528 

9533 

9538 

11.5 
13.8 

11.0 
13.2 

90 

9542 

5547 

9552 

3557 

3562 

9566 

9571 

9576 

9581 

9586 

8 

16.1 
18.4 

15.4 
17.6 

91 
92 

9590 
9638 

)595 
)643 

9600 
9047 

3605 
3652 

9609 
9657 

9614 
9661 

9619 
9666 

9624 
9671 

9628 
9675 

9633 
9680 

9 

20.7 

19.8 

93 

9685 

)689 

9694 

3699 

9703 

9708 

9713 

9717 

9722 

9727 

21 

2.1 
4.2 

94 

9731  < 

)736 

9741 

3745 

9750 

9754 

9759 

9763 

9768 

9773 

1 

95 

9777  < 

)782 

9786 

3791 

9795 

9800 

9805 

9809 

9814 

9818 

3 
4 
5 

6.3 
8.4 
10.5 

96 

9823  * 

)827 

9832 

3836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

)872 

9877 

3881 

9886 

9890 

9894 

9899 

9903 

9908 

6 

7 

12.6 
14.7 

98 

9912 

)917 

9921 

3926 

3930 

9934 

9939 

3943 

9948 

9952 

8 

16.8 

99 

9956 

)961 

9965 

3969 

9974 

9978 

9983 

3987 

9991 

9996 

9   18.9  1 

No. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Prop.  Parts 

Log,,, /I 
Log.  trig. 


330 


TABLE    II.     LOGARITHMIC  SINES,  COSINES, 


X 

log  sin 

d 

log  cos 

d 

log  tan 

d 

log  cot 

SmaU  Angles 

0°     0' 

—  00 

10.0000 

0 
0 
0 
0 

—  oc 

00 

90°   0' 

X 

«       1 

T       1 

10' 

7.4637 

3011 
1760 

.0000 

7.4637 

3011 
1761 

2.5363 

50'    < 

a°  6.4637  6.4637 

20' 

■;. .  7648 

.0000 

.7648 

.2352 

40' 

r  6.4637  6.4638 

30' 

.9408 

1250 

.0000 

.9409 

1249 

.0591 

30' 

2°  6.4636  6.4639 

40' 

8.0658 

969 
792 
669 
580 
511 
458 

.0000 

0 

1 

0 
0 
0 
1 

8.0658 

969 
792 
670 
580 
512 
457 

1.9342 

20' 

3°  6.4635  6.4641 

50' 
1"     0' 

10' 

.1627 

8.2419 

.3088 

.0000 

9.9999 

.9999 

.1627 

8.2419 

.3089 

.8373 

1.7581 

.6911 

10' 
89°   0' 

50' 

4°  6.4634  6.4644 
5°  6.4631  6.4649 

20' 

.3668 

.9999 

.3669 

.6331 

40' 

30' 

.4179 

9.9999 

.4181 

.5819 

30' 

40' 

50' 

2°     0' 

.4637 
.5050 

8.5428 

413 
378 
348 

.9998 

.9998 

9.9997 

0 

1 

0 

.4638 

.5053 

8.5431 

415 
378 
348 

.5362 

.4947 

1.4569 

2D' 

10' 

88°   0' 

Prop.  Parts. 

113  111 

109 

10.9 

21.8 

^j    10' 

.5776 

321 
300 
280 

.9997 

1 

0 

1 

.5779 

322 
300 

281 

.4221 

50' 

1  11.3    11.1 

2  22.6    22.2 

'  '20' 

.6097 

.9996 

.6101 

.3899 

40' 

3    33.9    33.3 

32.7 

30' 

.6397 

.9996 

.6401 

.3599 

30' 

4  45.2    44.4 

5  56.5    55.5 

43.6 
54.5 

10' 

.6677 

.9995 

0 

1 

.6682 

263 
249 

.3318 

20' 

6    67. J 

66.6 

65.4 

.      50' 

.6940 

263 

248 

.9995 

.6945 

.3055 

10' 

7  79. 

8  90.^ 

77.7 
88.8 

76.3 

87.2 

3°     0' 

:^  10' 

8.7188 

235 

9.9994 

1 

8.7194 

235 

1.2806 

87°  0' 

9  101.' 

99.9 

98.1 

.7423 

222 
212 
202 

.9993 

0 

1 
1 

.7429 

223 
213 
202 

.2571 

50' 

]  20' 
,'"  30' 

.7645 

.7857 

.9993 
.9992 

.7652 
.7865 

.2348 
.2135 

40' 
30'    " 

108 

107 

105 

'    40' 

.8059 

.9991 

I 

.8067 

194 
185 

.1933 

20' 

1    10. s 

10.7 

10.5 

",.  50' 

.8251 

192 
185 

.9990 

1 

.8261 

.1739 

10' 

2  21. f 

3  22.4 

21.4 

21.0 
31.5 

4°  .  0' 

8.8436 

177 
170 
163 

158 
152 
147 

9.9989 

0 

8.8446 

178 

1.1554 

86°  0' 

4    43.^ 

42^8 

42.0 

.  10' 
'  20' 

.8613 
.8783 

.9989 
.9988 

1 
1 

.8624 
.8795 

171 
165 

.1376 
.1205 

50' 
40' 

5  54. C 

6  64. S 

7  75. f 

53.5 
64.2 
74.9 

52.5 
63.0 
73.5 

30' 
40' 

.8946 
.9104 

.9987 
.9986 

1 

1 

.8960 
.9118 

158 
154 
.148 

.1040 

.0882 

30' 
20' 

8  86.4 

9  97.5 

85.6 
96.3 

84.0 
94.5 

50' 

.9256 

.9985 

2 

.9272 

.0728 

10' 

1 

5°     0' 

10' 
J     20' 

8.9403 
.9545 
.9682 

9.9983 
.9982 
.9981 

8:9420 
.9563 
.9701 

143 
138 
135 

1.0580 
.0437 
.0299 

85°  0' 

50' 
40' 

1 

142 
137 
134 
129 
125 

1 
1 

1 

104 

1  W.4 

2  20.5 

102 

10.2 
20.4 

101 

10.1 
20.2 

-'     30' 
40' 

.9816 
.9945 

.9980 
.9979 

1 

2 

.9836 
.9966 

130 
127 
123 
120 

.0164 
.0034 

30' 
20' 

3  31. i 

4  41. ( 

5  52. C 

30.6 
40.8 
51.0 

30.3 
40.4 
50.5 

50' 

9.0070 

.9977 

1 
1 

9.0093 

0.9907 

10' 

6    62.4 

61.2 

60.6 

6°     0' 

9.0192 

122 
119 

9.9976 

9.0216 

0.9784 

84°  0' 

7  72. S 

8  83.2 

71.4 
81.6 

70.7 
80.8 

10' 

.0311 

.9975 

2 
I 

.0336 

117 
114 

.9664 

50' 

9  93. e 

91.8 

90.9 

20' 

.0426 

115 
113 
109 
107 

.9973 

.0453 

.9547 

40' 

30' 
40' 

.0539 
.0648 

.9972 
.9971 

1 
2 

.0567 
.0678 

111 
108 
105 
104 

.9433 
.9322 

30' 
20'    - 

99 

98 

97 

95 

50' 

.0755 

.9969 

1 

.0786 

.9214 

10'    1 

9.9 

9.8 

9. 

7   9.5 

7°     0' 

9.0859 

104 
102 

9.9968 

2 

9.0891 

0.9109 

83°  0'    I 

19.8  1 
29.7  2 

9.6 
9.4 

19. 

29. 

4  19.0 

128.5 

10' 

.0961 

.9966 

2 

1 

.0995 

101 
98 

.9005 

50'    4 

39.6  3 

9.2 

38. 

8  38.0 

20' 

.1060 

99 
97 

.9964 

.1096 

.8904 

Z ' 

49.5  4 
59.4  5 

9.0 

S.8 

48. 
58. 

5  47.5 
2  57.0 

30' 

.1157 

.9963 

.1194 

.8806 

30'    7 

69.3  6 

8.6 

67. 

9  66.5 



8 

79.2  7 

8.4 

77. 

6  76.0 

log  cos 

d 

log  sin 

d 

log  cot 

d 

log  tan 

X        S 

89.1b 

8.2187 

3  85.5 

143 

142" 

13f 

i  137 

Y35 

134 

130 

129 

127 

125 

123 

122 

119 

117 

115 

lu 

1    14.3 

14.2 

13. 

i    13.7 

13.5 

13.4 

13.0 

12.9 

12.7 

12.5 

12.3 

12.2 

11.9 

11.7 

11.5 

11.4 

2    28.6 

28.4 

27. 

3    27.4 

27.0 

26.8 

26.0 

25.8 

25.4 

25.0 

24.6 

24.4 

23.8 

23.4 

23.0 

22.8 

3    42.9 

41. 

1    41.1 

40.5 

40.2 

38.7 

38.1 

37.5 

36.9 

36.6 

35.7 

35.1 

34.5 

34.2 

4    57.2 

56^8 

55. 

2    54.8 

54.0 

53.6 

52;o 

51.6 

50.8 

50.0 

49.2 

48.8 

47.6 

46.8 

46.0 

45.6 

5   71.5 

71.0 

69. 

D    68.5 

67.5 

67.0 

65.0 

61.5 

63.5 

62.5 

61.5 

61.0 

59.5 

58.5 

57.5 

57.0 

6   85.8 

85.2 

82. 

8    82.2 

81.0 

80.4 

78.0 

77.4 

76.2 

75.0 

73.8 

73.2 

71.4 

70.2 

69.0 

68.4 

7  100.1 

99.4 

96. 

6    95.9 

94.5 

93.8 

91.0 

90.3 

88.9 

87.5 

86.1 

85.4 

83.3 

81.9 

80.5 

79.8 

8  114.4  1 

13.6 

110. 

4  109.6 

08.0 

107.2 

104.0 

103.2 

101.6 

100.0 

98.4 

97.6 

95.2 

93.6 

92.0 

91.2 

9  128.7  1 

27.8 

124. 

2  123.3 

21.5 

120.6 

117.0 

116.1 

114.3 

112.5 

110.7 

109.8 

107.11105.31103.51102.6    1 

TANGENTS  AND  COTANGENTS.     TABLE  II 


331 


1  * 

log  sin 

d 

log  COS 

d 

log  tan 

d 

log  cot 

Prop.  Parts 

30 

9.1157 

95 

9.9963 

2 

9.1194 

97 
94 
93 

0.8806 

30' 

" 

40 

.1252 

.9961 

.1291 

.8709 

20' 

73 

71 

70 

69 

68 

50 

.13«i   - 

.9959 

2 
1 

.1385 

.8615 

10'  , 

7.3 
14.6 

7.1 
14.2 

7.C 
14. C 

6.S 
13. i 

6,8 
13.6 

8°  0 

9.1436 

89 

87 
85 

9.9958 

2 

2 

2 

9 

9.1478 

91 
89 

87 

0.8522 

82°  0'  ^ 

21.9 

21.3121.C 

20.7 

20.4 

10 
20 

.1525 
.1612 

.9956 
.9954 

.1569 
.1658 

.8431 
.8342 

50'  i 
40'  e 

29.2 
36.5 
43.8 

28.4 
35  5 
42.6 

28.0 
.35.0 
42  0 

27.6 
.34.5 
141.4 

27.2 
34.0 
40.8 

30 
40 

.1697 
.1781 

84 
82 
80 
79 
78 
76 

.9952 
.9950 

.1745 
.1831 

86 
84 
82 
81 
80 
78 

.8255 
.8169 

30'  ; 

20'  a 

51.1 
58  4 
65.7 

49.7 
56.8 
63.9 

49.0 
56.0 
63.0 

18.3 
55.2 
62.1 

47.6 
54  4 

:6i.2 

50 

.1863 

.9948 

2 
2 
2 
2 

.1915 

.8085 

10'  - 

1 

9°  0 

9.1943 

9.9946 

9.1997 

0.8003 

81°  0' 

10' 

.2022 

.9944 

.2078 

.7922 

50' 

67 

66 

66 

64 

63 

20' 

.2100 

.9942 

.2158 

.7842 

40'  \ 

6.7 
13.4 

6.6 
13.2 

6.5 
13.0 

6.4 
12.8 

6  3 
12.6 

30' 

.2176 

75 
73 
73 
71 
70 
68 
68 
66 
66 

.9940 

2 
2 
2 

3 
2 
2 
3 
2 
3 

.2236 

77 
76 
74 

.7764 

30'  3 

20.1 

19.8 

19.5 

19.2 

18.9 

40' 
50' 

.2251 
.2324 

.9938 
.9936 

.2313 
.2389 

.7687 
.7611 

20'  \ 
10'  6 

26.8 
33.5 
40.2 

26.4 
33.0 
39.6 

26.0 
39:c 

25.6 
32. ( 

25.2 
31.5 
37.8 

10°  0' 

10' 

9.2397 
.2468 

9.9934 
.9931 

9.2463 
.2536 

73 
73 
71 
70 
69 
68 

0.7537 
.7464 

80°  0'  I 

50'  1 

46.9 
53.6 
60.3 

46.2 
52.8 
59.4 

45.5 
52.0 
58.5 

U'.i 
51.2 
57.6 

44.1 
50.4 
56.7 

20' 

.2538 

.9929 

.2609 

.7391 

40'  - 



30' 

.2606 

.9927 

.2680 

.7320 

30' 

40' 

.2674 

.9924 

.2750 

.7250 

20' 

61 

60 

69 

68 

67 

50' 

.2740 

.9922 

.2819 

.7181 

10'  I 

6.1 

12.2 

6.0 
12.0 

5.9 
11.8 

5.8 
11.6 

5.7 
11.4 

11°  0' 

9.2806 

64 
64 
63 
61 
61 
60 
59 
58 
57 

9.9919 

2 
3 
2 
3 
2 
3 
3 

9.2887 

66 
67 
65 
64 
63 
63 
61 

0.7113 

79°  0'  3 

18.3 

18.0 

17.7 

17.4 

17.1 

10' 
20' 

.2870 
.2934 

.9917 
.9914 

.2953 
.3020 

.7047 
.6980 

50'  I 
40'  t 

24.4 
30.5 
36.6 

24.0 
30.0 
36.0 

23.6 
29.5 
35.4 

23.2 
29.0 
34.8 

22.8 
28.6 
34.2 

30' 
40' 

.2997 
.3058 

.9912 
.9909 

.3085 
.3149 

.6915 
.6851 

30'  I 
20'  I 

42.7 
48.8 
54.9 

42.0 
48.0 
54.0 

41.3 
47.2 
53.1 

40.6 
46.4 
52.2 

39.9 
45.6 
51.3 

50' 

.3119 

.9907 

.3212 

.6788 

10'  - 

_ 

12°  0' 

9.3179 

9.9904 

9.3275 

0.6725 

78°  0' 

10' 

.3238 

.9901 

.3336 

.6664 

50' 

66 

55 

54 

53 

52 

20' 

.3296 

.9899 

2 
3 

.3397 

61 
61 

.6603 

40'  1 

5.6 
11.2 

5.5 
11.0 

5.4 

10.8 

5.3 
10.6 

5.2 
10.4 

30' 

.3353 

57 

.9896 

3 

.3458 

59 

.6542 

30'  3 

16.8 

16.5 

16.2 

15.9 

15.6 

40' 

3410 

.9893 

.3517 

.6483 

20'  * 

22.4 

22.0 

21.6 

21.2 

20.8 

50' 

.3466 

56 
55 

.9890 

3 
3 

.3576 

59 
58 

.6424 

10'  1 

28.0 
33.6 

27.5 
33.0 

27.0 
32.4 

26.5 
31.8 

26.0 
31.^ 

13°  0' 

10' 

9.3521 
.3575 

54 
54 
53 
52 
52 

9.9887 
.9884 

3 
3 
3 
3 
3 

9.3634 
.3691 

57 
57 
56 
55 
55 

0.6366 
.6309 

77°  C  I 

50'  g' 

39.2 
44.8 
50.4 

38.5 
44.0 
19.5 

37.8 
43.2 
48.6 

37.1 
42.4 
47.7 

36.4 
41.6 
46.8 

20' 

.3629 

.9881 

.3748 

.6252 

40'  - 

30' 

.3682 

.9878 

.3804 

.6196 

30' 

40' 

.3734 

.9875 

.3859 

.6141 

20' 

51 

50 

48 

47 

50' 

.3786 

.9872 

.3914 

.6086 

10'  1 

5.1 

5.0 

4.8 

4.7 

51 

3 

54 

2 

10  2 

10.0 

9.6 

9.4 

14°  0' 

9.3837 

50 

9.9869 

3 

9.3968 

53 

0.6032 

76°  0'  3 

15.3 

15.0 

14.4 

14.1 

10' 

.3887 

.9866 

.4021 

.5979 

50'  4 

20.4 

20.0 

19.2 

18.8 

20' 

.3937 

50 
49 

.9863 

3 
4 

.4074 

53 
53 

.5926 

40'  I 

25.5 
30.6 

25.0 
30.0 

24.0 

23.5 
28.2 

30' 

.3986 

.9859 

.4127 

.5873 

30'  I 

35.7 

35.0 

33^6 

32.9 

40' 

.4035 

49 

48 
47 

.9856 

3 
3 
4 

.4178 

51 
52 
51 

.5822 

20' 

40.8 
45.9 

40.0 
15.5 

43]2 

37.6 
42.3 

50' 

.4083 

.9853 

.4230 

.5770 

10' 

15°  0' 

9.4130 

9 . 9849 

9.4281 

0.5719 

75°  0' 

log  cos 

d 

log  sin 

d  [log  cot| 

d 

log  tan 

X 

97 

94 

93 

91 

89 

87 

86 

86 

84 

82 

81 

79 

78 

77 

76 

76 

74 

1   9.7 

9.4 

9.3 

9.1 

8.9 

8.7 

8.6 

8.^ 

8.4 

8.2 

8.1 

7.9 

7. 

8  7. 

7  7.6 

7.5 

7  4 

2  19,4 

18.8 

18.6 

18.2 

17.8 

17.4 

17  2 

17. C 

16.8 

16.4 

16.2 

15.8 

15. 

6  15 

4  15.2 

5.0 

14.8 

3  29.1 

27.9 

27.3 

26.7 

26.1 

25.8 

25.  J 

25.2 

24.6 

24. C 

23.7 

23. 

4  23. 

1  22.8  5 

2.5 

22.2 

4  38.8 

37^6 

37.2 

36.4 

35.6 

34.8 

34.4 

34. C 

33.6 

32.8 

32.4 

31.6 

31. 

2  30. 

8  30.4  . 

!0.0 

29.6 

5  48.5 

47.0 

46.5 

45.5 

44.5 

43.5 

13.0 

42. J 

42.0 

41.0 

AO.t 

39.5 

39 

0  38. 

5  38.0  : 

7.5 

37.0 

6  58.2 

56.4 

55.8 

54.6 

53.4 

52.2 

51.6 

51. C 

50.4 

49.2 

48. e 

47.4 

46. 

8  46. 

2  45  6  4 

5.0 

44.4 

7  67.9 

65.8 

65.1 

63.7 

62.3 

60.9 

jO.2 

59. ^ 

58.8 

57.4 

56.- 

55.3 

54. 

6  53. 

9  53.2  . 

2  5 

51.8 

8  77.6 

75.2 

74.4 

72.8 

71.2 

68. C 

67.2 

65.6 

64. S 

63.2 

62. 

4  61. 

6  60.8  C 

)0.0 

59.2 

9  87.3 

84. 6 

83.7 

81.9 

80.1 

78^3 

77'4 

76.  £ 

75  6 

73.8  1  72. G 

71.1 

70. 

2  69. 

3  68.4  f 

7.5 

66.6 

332        TABLE  II.     LOGARITHMIC  SINES,  COSINES, 


X 

log  Bin 

d 

log  cos 

d 

log  tan 

d 

log  cot 

Prop.  Parts 

15°  0' 

9.4130 

47 
46 

9.9849 

3 
3 

9.4281 

50 
50 

0.5719 

75°  0' 

60 

49 

48 

47 

10' 

.4177 

.9846 

.4331 

.5669 

50' 

1 

5.0 

4.9 

4.8 

4.7 

20^ 

"  . 4223 

.9843 

.4381 

.5619 

40' 

2 

10.0 

9.8 

9.6 

9.4 

46 

4 

49 

3 

15,0 

14.7 

14.4 

14.1 

30' 

.4269 

.9839 

.4430 

.5570 

30' 

4 

20.0 

19.6 

19.2 

18.8 

40' 

.4314 

45 
45 

44 

.9836 

3 

4 
4 

.4479 

49 
48 
48 

.5521 

20' 

5 
6 

25.0 
30  0 

24.6 
29.4 

24.0 

28.8 

23.5 

28.2 

50' 

.4359 

.9832 

.4527 

.5473 

10' 

7 

35.0 

34.3 

33.6 

32.9 

8 

40.0 

39.2 

37.6 

16°  0' 

9.4403 

44 
44 
42 

9.9828 

3 

4 
4 

9.4575 

47 
47 
47 

0.5425 

74°  0' 

9 

45.0 

44.1 

43^2 

42.3 

10' 

.4447 

.9825 

.4622 

.5378 

50' 

20' 

.4491 

.9821 

.4669 

.5331 

40' 

30' 

.4533 

.9817 

.4716 

.5284 

30' 

40' 

.4576 

43 

.9814 

3 

.4762 

46 

.5238 

20' 

46^ 

45 

^ 

43 

50' 

.4618 

42 
41 

.9810 

4 

4 

.4808 

46 
45 

.5192 

10' 

1 
2 

4.6 
9.2 

4.5 
9.0 

4.4 

8.8 

4.3 
8.6 

17°  0' 

9.4659 

41 
41 

9.9806 

4 

9.4853 

45 
45 

0.5147 

73°  0' 

4 

13.8 

18.4 

13.5 
18.0 

13.2 
17.6 

12.9 

17.2 

10' 

.4700 

.9802 

4 

.4898 

.5102 

50' 

5 

23.0 

22.6 

22.0 

21,5 

20' 

.4741 

.9798 

.4943 

.5057 

40' 

6 

27.6 

27.0 

26.4 

25.8 

40 

4 

44 

7 

32.2 

31.6 

30.8 

30.1 

30' 

.4781 

.9794 

.4987 

.5013 

30' 

8 

36.8 

36.0 

35.2 

34.4 

40' 

.4821 

40 
40 
39 

.9790 

4 
4 
4 

.5031 

44 
44 
43 

.4959 

20' 

9 

41.4 

40.5 

39.6 

38.7 

50' 

.4861 

.9786 

.5075 

.4925 

10' 

18°  0' 

9.4900 

39 
38 
38 

9.9782 

4 
4 
4 

9.5118 

43 
42 
42 

0.4882 

72°  0' 

10' 

.4939 

.9778 

.5161 

.4839 

50' 

" 

^ 

41 

40 

39 

20' 

.4977 

.9774 

.5203 

.4797 

40' 

1 

4.2 

8.4 

l\ 

4.0 
8.0 

3.9 

7.8 

30' 

.5015 

37 
38 
36 

.9770 

5 
4 
4 

.5245 

42 
42 
41 

.4755 

30' 

3 

12.6 

12.3 

12.0 

11.7 

40' 

.5052 

.9765 

.5287 

.4713 

20' 

4 

16. S 

16.4 

16.0 

15.6 

50' 

.5090 

.9761 

.5329 

.4671 

10' 

5 
6 

21.0 

25.2 

20.5 
24.6 

20.0 
24.0 

19.5 
23.4 

19°  0' 

9.5126 

37 
36 
36 

9.9757 

5 

9.5370 

41 

0.4630 

71°  0' 

8 

29.4 
33.6 

ii 

28.0 
32.0 

27.3 
31.2 

10' 

.5163 

.9752 

4 
5 

.5411 

40 
40 

.4589 

50' 

9 

37.8 

36.9 

36.0 

35.1 

20' 

.5199 

.9748 

.5451 

.4549 

40' 

30' 

.5235 

35 
36 
35 

.9743 

4 
5 
4 

.5491 

40 
40 
40 

.4509 

30' 

40' 

.5270 

.9739 

.5531 

.4469 

•20' 

_ 

50' 

.5306 

.9734 

.5571 

.4429 

10' 

J 

38 

3..S 

37 

3.7 

36 

3.6 

35 

3.5 

20°  0' 

9.5341 

34 
34 
34 

34 

9.9730 

5 
4 
5 

5 
5 
4 

9.5611 

39 
39 
38 

39 
38 
38 

0.4389 

70°  0' 

2 

7.6 

7.4 

7.2 

7.0 

10' 
20' 

.5375 
.5409 

.9725 
.9721 

.5650 
.5689 

.4350 
.4311 

50' 
40' 

3 
4 
5 

11.4 
15.2 
19.0 

14:8 
18.5 

10.8 
14.4 
18.0 

10.5 
14.0 
17.6 

'  30' 

.5443 

.9716 

.5727 

.4273 

30' 

7 

22.8 
26.6 

22.2 
25.9 

21.6 
25.2 

21.0 
24.6 

40' 

.5477 

.9711 

.5766 

.4234 

20' 

8 

30.4 

29.6 

28.8 

28.0 

50' 

.5510 

33 
33 

.9706 

.5804 

.4196 

10' 

9 

34.2 

33.3 

32.4 

31.5 

21°  0' 

9.5543 

33 
33 

32 

9.9702 

5 
5 
5 

9.5842 

37 
38 
37 

0.4158 

69°  0' 

10' 

.5576 

.9697 

.5879 

.4121 

50' 

20' 

.5609 

.9692 

.5917 

.4083 

40' 

~ 

34 

W 

^, 

31 

30' 
40' 

.5641 
.5673 

32 
31 

.9687 
.9682 

5 
5 
5 

.5954 
.5991 

37 
37 
36 

.4046 
.4009 

30' 
20' 

3 

3.4 
6.8 
10.2 

3.3 
6.6 
9.9 

6:1 

9.6 

3.1 
6.2 
9.3 

50' 

.5704 

.9677 

.6028 

.3972 

10' 

4 

13.6 

12.8 

12.4 

32 

5 

17.0 

16.5 

16.0 

15.5 

22°  0' 

9.5736 

9.9672 

5 
6 
5 

9.6064 

36 
36 
36 

0.3936 

68°  0' 

6 

20.4 

19.8 

19.2 

18.6 

10' 
20' 

.5767 
.5798 

31 
31 
30 

.9667 
.9661 

.6100 
.6136 

.3900 
.3864 

50' 
40' 

7 
8 
9 

23.8 
27.2 
30.6 

23.1 
26.4 
29.7 

22.4 
25.6 

28.8 

21.7 
24.8 
27.9 

30' 

.5828 

.9656 

.6172 

.3828 

30' 

jlog  cos 

d 

log  sin 

I 

log  cot 

d 

log  tan 

X 

Prop.  Parts 

TANGENTS  AND  COTANGENTS.     TABLE  II 


333 


X 

log  sin 

d 

log  cos 

d 

logtaa 

d 

log  cot 

Prop.  Parts 

30' 

9.5828 

31 
30 
30 

9.9656 

5 
5 
6 

9.6172 

36 
35 

0.3828 

30' 

36 

36 

34 

40' 

.5859 

.9651 

.6208 

.3792 

20' 

I 

3.6 

3.5 

3  4 

50' 

.5889 

.9646 

.6243 

.3757 

10' 

2 

7.2 

7.0 

6  ,S 

36 

3 

10  8 

10.5 

10  2 

23°   0' 

9.5919 

29 
30 
29 

29 
29 

28 

9.9640 

5 
6 
5 

6 
5 
6 

9.6279 

35 
34 
35 

34 
35 
34 

0.3721 

67°  0' 

4 

14.4 

14.0 

13  6 

10' 
20' 

.5948 
.5978 

.9635 
.9629 

.6314 
.6348 

.3686 
.3652 

50' 
40' 

5 

6 
7 

18.0 
21.6 
25.2 

17  5 
21  0 
24.5 

17  0 
20  4 

23  S 

30' 

.6007 

.9624 

.6383 

.3617 

30' 

8 
9 

28.8 
32.4 

28.0 
31.5 

27  2 
30  (\ 

40' 

.6036 

.9618 

.6417 

.3583 

20' 

50' 

.6065 

.9613 

.6452 

.3548 

10' 

24°   0' 

9.6093 

28 
28 
28 

9.9607 

5 
6 
6 

9.6486 

34 
33 
34 

0.3514 

66°  0' 

10' 

.6121 

.9602 

.6520 

.3480 

50' 

33 

"32^ 

31 

20' 

.6149 

.9596 

.6553 

.3447 

40' 

1 
2 

3.3 
6.6 

3.2 
6.4 

u 

30' 
40' 

.6177 
.6205 

28 
27 

.9590 
.9584 

6 
5 
6 

.6587 
.6620 

33 
34 
33 

.3413 
.3380 

30' 
20' 

3 
4 
5 

9.9 
13.2 
16.5 

9.6 
12.8 
16.0 

9  3 
12  4 
15  .3 

50' 

.6232 

27 

.9579 

.6654 

.3346 

10' 

6 

? 

19.8 
23.1 

19.2 
22.4 

18.6 
21  7 

25°   0' 

9.6259 

27 
27 

27 

9.9573 

6 
6 
6 

9.6687 

33 
32 
33 

0.3313 

66°  0' 

8 

26.4 

25.6 

24  8 

10' 

.6286 

.9567 

.6720 

.3280 

50' 

9 

29.7 

28.8 

27  9 

20' 

.6313 

.9561 

.6752 

.3248 

40' 

30' 

.6340 

26 
26 
26 

.9555 

6 
6 
6 

.6785 

32 
33 
32 

.3215 

30' 

40' 

.6366 

.9549 

.681.7 

.3183 

20' 

— 

30 

~29 

28 

50' 

.6392 

.9543 

.6850 

.3150 

10' 

1 

3.0 

2.9 

2.8 

26°   0' 

9.6418 

26 
26 
25 

9.9537 

7 
6 
6 

9.6882 

32 
32 
31 

0.3118 

64°  0' 

2 
3 

6.0 
9.0 

5.8 
8.7 

5  0 
8.4 

10' 

.6444 

.9530 

.6914 

.3086 

50' 

4 

12.0 

11.6 

11  2 

20' 

.6470 

.9524 

.6946 

.3054 

40' 

5 
6 

15.0 
18.0 

14.5 
17.4 

14  0 

16  8 

30' 
40' 

.6495 
.6521 

26 

25 
24 

.9518 
.9512 

6 

.6977 
.7009 

32 
31 
32 

.3023 
.2991 

30' 
20' 

7 
8 
9 

21  0 
24.0 
27.0 

20.3 
23.2 
26.1 

19  6 
22  4 

25.2 

50' 

.6546 

.9505 

.7040 

.2960 

10' 

27°   0' 

9.6570 

25 
25 
24 

9.9499 

9.7072 

31 
31 
31 

0.2928 

63°  0' 

10' 

.6595 

.9492 

.7103 

.2897 

50' 

20' 

.6620 

.9486 

.7134 

.2866 

40' 

1 

27 

2.7 

"26" 

25 

2.5 

30' 

.6644 

24 
24 
24 

.9479 

.7165 

31 
30 
31 

.2835 

30' 

2 

5.4 

5  2 

50 

40' 
50' 

.6668 
.6692 

.9473 
.9466 

.7196 
.7226 

.2804 
.2774 

20' 
10' 

3 
4 
5 

8.1 
10.8 
13.5 

7.8 
10  4 
13  0 

7.5 
10.0 
12.5 

28°   0' 

9.6716 

24 
23 
24 

9.9459 

9.7257 

30 
30 
31 

0.2743 

62°  0' 

6 
7 

16.2 
18.9 

15  6 
18  2 

150 
17  5 

10' 

.6740 

.9453 

.7287 

.2713        50' 

8 

21.6 

20  8 

20  0 

20' 

.6763 

.9446 

.7317 

.2683 

40' 

9 

24.3 

23.4 

22.5 

30' 

.6787 

23 
23 
23 

.9439 

.7348 

30 
30 
30 

.2652 

30' 

40' 

.6810 

.9432 

.7378 

.2622 

20' 

50' 

.6833 

.9425 

.7408 

.2592 

10' 

— ■ 

2r 

23" 

"22 

29°  0' 

10' 

9.6856 

.6878 

22 
23 
22 

9.9418 
.9411 

9.7438 
.7467 

29 
30 
29 

0.2562 
.2533 

61°  0' 

50' 

1 
2 
3 

2.4 

4.8 
7  2 

2.3 
4.6 
6  y 

4  4 

0  6 

20' 

.6901 

.9404 

.7497 

.2503 

40' 

4 
5 

9  6 
12.0 

9  2 
U  5 

8.8 
11.0 

30' 

.6923 

23 
22 
22 

.9397 

8 

.7526 

30 
29 
29 

.2474 

30' 

6 

14  4 

13  8 

13  2 

40' 
50' 

.6946 
.6968 

.9390 
.9383 

.7556 
.7585 

.2444 
.2415 

20' 
10' 

7 
8 
9 

16  8 
19.2 
21.6 

16.1 

18  4 
20.7 

154 

17  6 
19.8 

30°  0' 

9.6990 

9.9375 

9.7614 

0.2386 

60°  C 

log  cos 

d 

log  sin 

d 

log  cot 

d 

log  tan 

X 

Prop.  Parts 

334 


TABLE  II.     LOGARITHMIC  SINES,  COSINES, 


X 

log  sin 

d 

log  cos 

d 

log  tan 

d 

log  cot 

Prop.  Parts 

30°  0' 

9.6990 

22 

9.9375 

7 

9.7614 

30 

0.2386 

60°  0' 

30 

3  0 

29 

2  9 

28 

2  8 

10' 

.7012 

21 

.9368 

7 

.7644 

29 

.2356 

50' 

2 

6^0 

5.8 

5.6 

20' 

.7033 

22 

.9361 

8 

.7673 

28 

.2327 

40' 

9.0 
12.0 

8.7 
11.6 

8.4 
11.2 

30' 

.7055 

21 

.9353 

7 

.7701 

29 

.2299 

30' 

15.0 

14.5 

14.0 

40' 

.7076 

21 

.9346 

8 

.7730 

29 

.2270 

20' 

18.0 
21  0 

17.4 
20  3 

16.8 
19  6 

50' 

.7097 

21 

.9338 

7 

.7759 

29 

.2241 

10' 

8 
9 

24.0 
27.0 

23.2 
26.1 

22.4 
25.2 

31°  0' 

9.7118 

21 

9.9331 

8 

9.7788 

28 

0.2212 

59°  0' 

10' 

.7139 

21 

.9323 

8 

.7816 

29 

.2184 

50' 

20' 

.7160 

21 

.9315 

7 

.7845 

28 

.2155 

40' 

30' 

.7181 

20 

.9308 

g 

.7873 

29 

.2127 

30' 

40' 

.7201 

21 

'9300 

8 

^7902 

28 

.2098 

20' 

27 

26 

"22" 

50' 

.7222 

20 

.9292 

8 

.7930 

28 

.2070 

10' 

1 
2 

2.7 
5.4 

2.6 
5.2 

2,2 
4.4 

32°  0' 

9.7242 

20 

9.9284 

■8 

9.7958 

28 

0.2042 

58°  0' 

3 

4 

8.1 

10  8 

7.8 
10  4 

6.6 

8  8 

10' 

.7262 

20 

.9276 

8 

.7986 

28 

.2014 

50' 

5 

13^5 

13:0 

11.0 

20' 

.7282 

20 

.9268 

8 

.8014 

28 

.1986 

40' 

6 

7 

16.2 
18.9 

15.6 
18.2 

13.2 
15.4 

30' 

.7302 

20 

.9260 

8 

.8042 

28 

.1958 

30' 

8 

21.6 

20,8 

17.6 

40' 

.7322 

20 

.9252 

8 

.8070 

27 

.1930 

20' 

9 

24.3 

23.4 

19.8 

50' 

.7342 

19 

.9244 

8 

.8097 

28 

.1903 

10' 

33°  0' 

9.7361 

\l 

9.9236 

8. 

9.8125 

28 

0.1875 

57°  0' 

10' 

.7380 

20 

.9228 

9 

.8153 

27 

.1847 

50' 

21 

20 

19 

20' 

.7400 

19 

.9219 

8 

.8180 

28 

.1820 

40' 

1 

2.1 
4.2 

2:0 
4.0 

1,9 

3,8 

30' 

.7419 

19 

.9211 

8 

.8208 

27 

.1792 

30' 

3 

6.3 

6.0 

5.7 

40' 

.7438 

19 

.9203 

9 

.8235 

28 

.1765 

20' 

4 

8.4 

8.0 

7.6 

50' 

.7457 

19 

.9194 

8 

.8263 

27 

.1737 

10' 

5 

6 

10.5 
12.6 

10.0 
12.0 

9.5 
11.4 

34°  0' 

9.7476 

18 

9.9186 

9 

9.8290 

•  27 

0.1710 

56°  0' 

7 
g 

14.7 
16  8 

14.0 
16  0 

13.3 
15  2 

10' 

.7494 

19 

.9177 

8 

.8317 

27 

.1683 

50' 

9 

18^9 

18!0 

17^1 

20' 

.7513 

18 

.9169 

9 

.8344 

27 

.1656 

40' 

30' 

.7531 

19 

.9160 

9 

.8371 

27 

.1629 

30' 

40' 

.7550 

18 

.9151 

9 

.8398 

27 

.1602 

20' 

50' 

^7568 

18 

'9142 

8 

^8425 

27 

.1575 

■  10' 

J 

l8 

1.8 

3.6 

17 

1,7 
3,4 

16 

1  6 

35°  0' 

9.7586 

18 

9.9134 

9 

9.8452 

27 

0.1548 

55°  0' 

2 

3.2 

10' 

.7604 

18 

.9125 

9 

.8479 

27 

.1521 

50' 

3 
4 
5 

5.4 
7.2 
9.0 

5,1 

6.8 
8,5 

4.8 
6.4 
8.0 

20' 

.7622 

18 

.9116 

9 

.8506 

27 

.1494 

40' 

30' 

.7640 

17 

.9107 

9 

.8533 

26 

.1467 

30' 

6 
7 

10.8 
12.6 

10,2 
11.9 

9.6 
11.2 

40' 

.7657 

18 

.9098 

9 

.8559 

27 

.1441 

20' 

8 

14.4 

13.6 

12.8 

50' 

.7675 

17 

.9089 

9 

.8586 

27 

.1414 

10' 

9 

16.2 

15.3 

14.4 

36°  0' 

9.7692 

18 

9.9080 

10 

9.8613 

26 

0.1387 

54°  0' 

10' 

.7710 

17 

.9070 

9 

.8639 

27 

.1361 

50' 

20' 

.7727 

17 

.9061 

9 

.8666 

26 

.1334 

40' 

~9 

8 

^" 

30' 

.7744 

17 

.9052 

10 

.8692 

26 

.1308 

30' 

2 
3 

.9 
1.8 

2.7 

.8 
1.6 
2.4 

.7 
1  4 

40' 

.7761 

17 

.9042 

9 

.8718 

27 

.1282 

20' 

2^1 

50' 

.7778 

17 

.9033 

10 

.8745 

26 

.1255 

10' 

4 
5 
6 

3.6 
4.5 
5.4 

3.2 
4,0 

4.8 

2.8 
3.5 
4.2 

37°  0' 

9.7795 

16 

9.9023 

9 

9.8771 

26 

0.1229 

53°  0' 

10' 

.7811 

17 

.9014 

10 

.8797 

27 

.1203 

50' 

7 
8 
9 

6.3 
7.2 
8.1 

5,6 

6,4 

7.2 

4.9 
5.6 
6.3 

20' 

.7828 

16 

.9004 

9 

.8824 

26 

.1176 

40' 

30' 

.7844 

.8995 

.8850 

,1150 

30' 

log  cos 

d 

log  sin 

d 

log  cot 

d 

log  tan 

X 

Prop.  Parts   | 

TANGENTS  AND  COTANGENTS.     TABLE  II 


335 


X 

log  sin 

d 

log  cos 

d  1 

ogtan 

d 

log  cot 

Prop.  Parts 

30' 

9.7844 

17 
16 
16 

9.8995 

10 
10 
10 

J.8850 

26 
26 
26 

0.1150 

30' 

40' 

.7861 

.8985 

.8876 

.1124 

20' 

50' 

.7877 

.8975 

.8902 

.1098 

10' 

38°  0' 

9.7893 

17 
16 
15 

16 

9.8965 

10 
10 
10 

10 
10 
10 

9.8928 

.26 
26 
26 

26 
26 
26 

0.1072 

52°  0' 

10' 
20' 

30' 

.7910 
.7926 

.7941 

.8955 
.8945 

.8935 

.8954 
.8980 

.9006 

.1046 
.1020 

.0994 

50' 
40'  ' 

30' 

26 

1  2.6 

2  5.2 

26 

2.5 
50 

40' 

.7957 

.8925 

.9032 

.0968 

20' 

3   7.8 

7.6 

50' 

.7973 

16 
16 

.8915 

.9058 

.0942 

10' 

4  10.4 

5  13.0 

10.0 
12.5 

39^  0' 

9.7989 

15 
16 
15 

9.8905 

10 

11 

10 

9.9084 

26 

0.0916 

51°  0' 

6  15.6 

7  18  2 

15.0 
17  5 

10' 

.8004 

.8895 

.9110 

25 
26 

.0890 

50' 

8  20^8 

2o:o 

20' 

.8020 

.8884 

.9135 

.0865 

40' 

9  23.4  22.5 

30' 

.8035 

15 
16 
15 

.8874 

10 

11 

10 

11 
11 
11 

10 

11 
11 

.9161 

26 
25 
26 

26 
25 
26 

26 
25 
26 

.0839 

30' 

40' 

.8050 

.8864 

.9187 

.0813 

20' 

50' 
40°  0' 

.8066 
9.8081 

.8853 
9.8843 

.9212 
9.9238 

.0788 
0.0762 

10' 
50°  0'  " 

17 

16 

15 

10' 
20' 

30' 

.8096 
.8111 

.8125 

15 
15 
14 

15 

.8832 
.8821 

.8810 

.9264 
.9289 

.9315 

.0736 
.0711 

.0685 

50' 
40' 

30' 

1  1.7 

2  3.4 

3  5.1 

4  6.8 

5  8.5 

1.6 
3.2 
4.8 
6.4 
8.0 

1.5 
3.0 
4.5 
6.0 
7.5 

40' 

.8140 

.8800 

.9341 

.0659 

20' 

6  10.2 

9.6 

9  0 

50' 

.8155 

15 
14 

.8789 

.9366 

.0634 

10' 

7  11.9 

8  13.6 

11.2 
12.8 

10.5 
12  0 

41°  0' 

9.8169 

15 
14 
15 

9.8778 

11 
11 
11 

9.9392 

25 
26 
25 

0.0608 

49°  0' 

9  15.3  14.4 

13  5 

10' 

.8184 

.8767 

.9417 

.0583 

50' 

20' 

.8198 

.8756 

.9443 

.0557 

40' 

30' 
40' 

.8213 

.8227 

14 
14 
14 

.8745 
.8733 

12 

11 
11 

.9468 
.9494 

26 
25 
25 

.0532 
.0506 

30' 
20' 

14 

13 

12 

50' 

.8241 

.8722 

.9519 

.0481 

10' 

1  1.4 

2  2.8 

1.3 
2.6 

1.2 
2.4 

42°  0' 

9.8255 

9.8711 

12 
11 
12 

11 
12 
12 

9.9544 

26 
25 
26 

25 
25 
26 

0.0456 

48°  0' 

3  4.2 

3.9 

3.6 

10' 
20' 

30' 

.8269 
.8283 

.8297 

14 
14 
14 

14 
13 
14 

.8699 
.8688 

.8676 

.9570 
.9595 

.9621 

.0430 
.0405 

.0379 

50' 
40' 

30' 

4  5.6 

5  7  0 

6  8.4 

7  9.8 

8  11.2 

5.2 
6.5 
7.8 
9.1 
10.4 

4.8 
6.0 
7.2 
8.4 
9.6 

40' 

.8311 

.8665 

.9646 

.0354 

20' 

9  12.6 

11.7 

10.8 

50' 

.8324 

.8653 

.9671 

.0329 

10' 

43°  0' 

9.8338 

13 
14 

9.8641 

12 
11 

9.9697 

25 
25 

0.0303 

47°  0' 

10' 

.8351 

.8629 

.9722 

.0278 
.0253 

50' 
40' 

20' 

.8365 

13 

.8618 

12 

.9747 

25 

11 

10 

30' 
40' 

.8378 
.8391 

13 
14 
13 

.8606 
.8594 

12 
12 
13 

.9772 
.9798 

26 
25 
25 

.0228 
.0202 

30' 
20' 

1  1.1 

2  2.2 

3  3.3 

1.0 
2.0 
3.0 

50' 

.8405 

.8582 

.9823 

.0177 

10' 

4  4.4 

5  5.5 

4.0 
5  0 

44°  0 

9.8418 

13 
13 
13 

9.8569 

12 
12 
13 

9.9848 

26 
25 
25 

0.0152 

46°  0' 

6  6.6 

6.0 

10 
20 

.8431 
.8444 

.8557 
.8545 

.9874 
.9899 

.0126 
.0101 

50' 
40' 

7  7.7 

8  8.8 

9  9.9 

7.0 
8.0 
9.0 

30 

.8457 

12 
13 
13 

.8532 

12 
13 
12 

.9924 

25 
26 
25 

.0076 

30' 

40 

.8469 

.8520 

.9949 

.0051 

20' 

50 

.8482 

.8507 

.9975 

.0025 

10' 

45°  0 

9.8495 

9.849£ 

0.0000 

0.0000 

45°  0' 

|lOg  C08 

d 

log  sinj  d 

log  cot 

d 

log  tan 

X 

Prop.  Pai 

ts 

336 


TABLE  III.     NATURAL  FUNCTIONS 


X 

sin  X 

COS  X 

.... 

cot  X 

sec  X 

cosec  X 

0°  0' 

10' 
20' 

.00000 
.00291 
.00582 

1.0000 
1.0000 
1.0000 

.00000 
.00291 
.00582 

cc 
343.77 
171.88 

1.0000 
1.0000 
1.0000 

00 

343.78. 
171.89 

90°  0' 

50' 
40' 

30' 
40' 
50' 

.00873 
.01164 
.01454 

1.0000 
.9999 
.9999 

.00873 
.01164 
.01455 

114.59 
85.940 
68.750 

1.0000 
1.0001 
1.0001 

114.59 
85.946 
68.757 

30' 
20' 
10' 

1°  0' 

10' 
20' 

.01745 
.02036 
.02327 

.9998 
.9998 
.9997 

.01746 
.02036 
.02328 

57.290 
49.104 
42^964 

1.0002 
1.0002 
1.0003 

57.299 
49.114 
42.976 

89°  0' 

50' 
40' 

30' 
40' 
50' 

.02618 
.02908 
.03199 

.9997 
.9996 
.9995 

.02619 
.02910 
.03201 

38.188 
34.368 
31.242 

1,0003 
1.0004 
1.0005 

38.202 
34.382 
31.258 

30' 
20' 
10' 

2°     0' 

10' 
20' 

.03490 
.03781 
.04071 

.9994 
.9993 
.9992 

.03492 
.03783 
.04075 

28.6363 
26.4316 
24.5418 

1.0006 
1.0007 
1.0008 

28.654 
26.451 
24.562 

88°  0' 

50' 
40' 

30' 
40' 
50' 

.04362 
.04653 
.04943 

.9990 
.9989 
.9988 

.04366 
.04658 
.04949 

22.9038 
21.4704 
20.2056 

1.0010 
1.0011 
1.0012 

22.926 
21.494 
20 . 230 

30' 
20' 
10' 

3°  0' 

10' 
20' 

.05234 
.05524 
.05814 

.9986 
.9985 
.9983 

.05241 
.05533 
.05824 

19.0811 
18.0750 
17.1693 

1.0014 
1.0015 
1.0017 

19.107 
18.103 
17.198 

87°  0' 

,  50' 
40' 

30' 
40' 
50' 

.06105 
.06395 
.06685 

.9981 
.9980 
.9978 

.06116 
.06408 
.06700 

16.3499 
15.6048 
14.9244 

1.0019 
1.0021 
1.0022 

16.380 
15.637 
14.958 

30' 
20' 
10' 

4°  0' 
10' 

20' 

.06976 
.07266 
.07556 

.9976 
.9974 
.9971 

.06993 
.07285 
.07578 

14.3007 
13.7267 
13.1969 

1.0024 
1.0027 
1.0029 

14.336 
13.763 
13.235 

86°  0' 

50' 
-  40' 

30' 
40' 
50' 

.07846 
.08136 
.08426 

.9969 
.9967 
.9964 

.07870 
.08163 
.08456 

12.7062 
12.2505 
11.8262 

1.0031 
1.0033 
1.0036 

12.746 
12.291 
11.868 

30' 
20' 
10' 

5°  0' 

10' 
20' 

.08716 
.09005 
.09295 

.9962 
.9959 
.9957 

.08749 
.09042 
.09335 

11.4301 
11.0594 
10.7119 

1.0038 
1.0041 
1.0044 

11.474 
11.105 
10.758 

85°  0' 

50' 
40' 

30' 
40' 
50' 

.09585 
.09874 
.10164 

.9954 
.9951 
.9948 

.09629 
.09923 
.10216 

10.3854 
10.0780 

9.7882 

1.0046 
1.0049 
1.0052 

10.433 
10.128 
9.839 

30' 
20' 
10' 

6°  0' 

10' 
20' 

.10453 
.10742 
.11031 

.9945 
.9942 
.9939 

.10510 
.10805 
.11099 

9.5144 
9.2553 
9.0098 

1.0055 
1.0058 
1.0061 

9.5668 
9.3092 
9.0652 

84°  0' 

50' 
40' 

30' 
40' 
50' 

.11320 
.11609 
.11898 

.9936 
.9932 
.9929 

.11394 
.11688 
.11983 

8.7769 
8.5555 
8.3450 

1.0065 
1.0068 
1.0072 

8.8337 
8.6138 
8.4647 

30' 
20' 
10' 

7°  0' 

10' 
20' 

.12187 
.12476 
.12764 

.9925 
.9922 
.9918 

.12278 
.12574 
.12869 

8.1443 
7.9530 
7.7704 

1.0075 
1.0079 
1.0083 

8.2055 
8.0157 
7.8344 

83°  0' 

50' 
40' 

30' 

.13053 

.9914 

.13165 

7.5958 

1.0086 

7.6613 

30' 

cos  X 

sin  X 

cot  X 

tan  X 

cosec  X 

sec  X 

X 

NATURAL  FUNCTIONS.     TABLE  III 


337 


X 

sinx 

COS  X 

tan  X 

COl  A 

BOO  X 

cosec  X 

30' 
40' 
50' 

.1305 
.1334 
.1363 

.9914 
.9911 
.9907 

.1317 
.1346 
.1376 

7.5958 
7.4287 
7.2687 

1.0086 
1.0090 
1.0094 

7.6613 
7,4957 
7.3372 

30' 
20' 
10' 

8°  0' 

10' 
20' 

.1392 
.1421 
.1449 

.9903 
.9899 
.9894 

.1405 
.1435 
.1465 

7.1154 
6.9682 
6.8269 

1.0098 
1.0102 
1.0107 

7.1853 
7.0396 
6.8998 

82°  0' 

50' 
40' 

30' 
40' 
50' 

.1478 
.1507 
.1536 

.9890 
.9886 
.9881 

.1495 
.1524 
.1554 

6.6912 
6.5606 
6.4348 

1.0111 
1.0116 
1.0120 

6.7655 
6.6363 
6.5121 

30' 
20' 
10' 

9°  0' 

10' 
20' 

.1564 
.1593 
.1622 

.9877 
.9872 
.9868 

.1584 
.1614 
.1644 

6.3138 
6.1970 
6.0844 

1.0125 
1.0129 
1.0134 

6.3925 
6.2772 
6.1661 

81°  0' 

50' 
40' 

30' 
40' 
50' 

.1650 
.1679 
.1708 

.9863 
.9858 
.9853 

.1673 
.1703 
.1733 

5.9758 
5.8708 
5.7694 

1.0139 
1.0144 
1.0149 

6.0589 
5.9554 
5.8554 

30' 
20' 
10' 

10°  0' 

10' 
20' 

.1736 
.1765 
.1794 

.9848 
.9843 
.9838 

.1763 
.1793 
.1823 

5.6713 
5.5764 
5.4845 

1.0154 
1.0160 
1.0165 

5.7588 
5.6653 
5.5749 

80°  0' 

50' 
40' 

30' 
40' 
50' 

.1822 
.1851 
.1880 

.9833 
.9827 
.9822 

.1853 
.1883 
.1914 

5.3955 
5.3093 
5.2257 

1.0170 
1.0176 
1.0182 

5.4874 
5.4026 
5.3205 

30' 
20' 
10' 

11°  0' 

10' 
20' 

.1908 
.1937 
.1965 

.9816 
.9811 
.9805 

.1944 
.1974 
.2004 

5.1446 
5.0658 
4.9894 

1.0187 
1.0193 
1.0199 

5.2408 
5.1636 
5.0886 

79°  0' 

50' 
40' 

30' 
40' 
50' 

.1994 
.2022 
.2051 

.9799 
.9793 
.9787 

.2035 
.2065 
.2095 

4.9152 
4.8430 
4.7729 

1.0205 
1.0211 
1.0217 

5.0159 
4.9452 
4.8765 

30' 
20' 
10' 

12°  0' 

10' 
20' 

.2079 
.2108 
.2136 

.9781 
.9775 
.9769 

.2126 
.2156 
.2186 

4.7046 
4.6382 
4.5736 

1.0223 
1.0230 
1.0236 

4.8097 
4.7448 
4.6817 

78°  0' 

50' 
40' 

30' 
40' 
50' 

.2164 
.2193 
.2221 

.9763 
.9757 
.9750 

.2217 
.2247 

.2278 

4.5107 
4.4494 
4.3897 

1.0243 
1.0249 
1,0256 

4.6202 
4.5604 
4.5022 

30' 
20' 
10' 

13°  0' 

10' 
20' 

.2250 
.2278 
.2306 

.9744 
.9737 
.9730 

.2309 
.2339 
.2370 

4.3315 
4.2747 
4.2193 

1,0263 
1.0270 
1.0277 

4.4454 
4.3901 
4.3362 

77°  0' 

50' 
40' 

30' 
40' 
50' 

.2334 
.2363 
.2391 

.9724 
.9717 
.9710 

.2401 
.2432 
.2462 

4.1653 
4.1126 
4,0611 

1.0284 
1.0291 
1.0299 

4.2837 
4.2324 
4.1824 

30' 
20' 
10' 

14°  0' 

10' 
20' 

.2419 
.2447 
.2476 

.9703 
.9696 
.9689 

.2493 
.2524 
.2555 

4.0108 
3.9617 
3.9136 

1.0306 
1.0314 
1.0321 

4.1336 
4.0859 
4.0394 

76°  0' 

50' 
40' 

30' 
40' 
50' 

.2504 
.2532 
.2560 

.9681 
.9674 
.9667 

.2586 
.2617 
.2648 

3. 8607 
3.8208 
3.7760 

1.0329 
1,0337 
1.0345 

3.9939 
3.9495 
3.9061 

30' 
20' 
10' 

15°  0' 

.2588 

.9659 

.2679 

3.7321 

1.0353 

3.8637 

75°  0' 

COB  A- 

sin  X 

cot  X 

tan  X 

coseo  X 

sec  X 

X 

338- 


TABLE  III.     NATURAL  FUNCTIONS 


X 

8in;c 

COS  X 

tan  X 

cot  X 

sec  X 

cosec  X 

15°  0' 

10' 
20' 

.2588 
.2616 
.2644 

.9659 
.9652 
.9644 

.2679 
.2711 
.2742 

3.7321 
3.6891 
3.6470 

1.0353 
1.0361 
1.0369 

3.8637 
3.8222 
3.7817 

75°  0' 

50' 
40' 

30' 
40' 
50' 

.2672 
.2700 
.2728 

.9636 
.9628 
.9621 

.2773 
.2805 
.2836 

3.6059 
3.5656 
3.5261 

1.0377 
1.0386 
1.0394 

3.7420 
3.7032 
3.6652 

30' 
20' 
10' 

16°  0' 

10' 
20' 

^56 

.2784 
.2812 

.9613 
.9605 
.9596 

.2867 
.2899 
.2931 

3.4874 
3.4495 
3.4124 

1.0403 
1.0412 
1.0421 

3.6280 
3.5915 
3.5559 

74°  0' 

50' 
40' 

30' 
40' 
50' 

.2840 
.2868 
.2896 

.9588 
.9580 
.9572 

.2962 
.2994 
.3026 

3.3759 
3.3402 
3.3052 

1.0430 
1.0439 
1.0448 

3.5209 
3.4867 
3.4532 

30' 
20' 
10' 

17°  0' 

10' 
20' 

.2924 
.2952 
.2979 

.9563 
.9555 
.9546 

.3057 
.3089 
.3121 

3.2709 
3.2371 
3.2041 

1.0457 
1.0466 
1.0476 

3.4203 
3.3881 
3.3565 

73°  0' 

50' 
40' 

30' 
40' 
50' 

.3007 
.3035 
.3062 

.9537 
.9528 
.9520 

.3153 
.3185 
.3217 

3.1716 
3.1397 
3.1084 

1.0485 
1.0495 
1.0505 

3.3255 
3.2951 
3.2653 

30' 
20' 
10' 

18°  0' 

10' 
20' 

.3090 
.3118 
.3145 

.9511 
.9502 
.9492 

.3249 
.3281 
.3314 

3.0777 
3.0475 
3.0178 

1.0515 
1.0525 
1.0535 

3.2361 
3.2074 
3.1792 

72°  C 

50' 
40' 

30' 
40' 
50' 

.3173 
.3201 
.3228 

.9483 
.9474 
.9465 

.3346 
.3378 
.3411 

2.9887 
2.9600 
2.9319 

1.0545 
1.0555 
1.0566 

3.1516 
3.1244 
3.0977 

30' 
20' 
10' 

19°  0' 

10' 
20' 

.3256 
.3283 
.3311 

.9455 
.9446 
.9436 

.3443 
.3476 
.3508 

2.9042 
2.8770 
2.8502 

1.0576 
1.0587 
1.0598 

3.0716 
3.0458 
3.0206 

71°  0' 

50' 
40' 

30' 
40' 
50' 

.3338 
.3365 
.3393 

.9426 
.9417 
.9407 

.3541 
.3574 
.3607 

2.8239 
2.7980 
2.7725 

1.0609 
1.0620 
1.0631  ■ 

2.9957 
2.9714 
2.9474 

30' 
20' 
10' 

20°  0' 

10' 
20' 

.3420 
.3448 
.3475 

.9397 
.9387 
.9377 

.3640 
.3673 
.3706 

2.7475 
2.7228 
2.6985 

1.0642 
1.0653 
1.0665 

2.9238 
2.9006 
2.8779 

70°  0' 

50' 
40' 

30' 
40' 
50' 

.3502 
.3529 
.3557 

.9367 
.9356 
.9346 

.3739 
.3772 
.3805 

2.6746 
2.6511 
2.6279 

1.0676 
1.0688 
1.0700 

2.8555 
2.8334 
2.8118 

30' 
20' 
10' 

21°   0' 

10' 
20' 

.3584 
.3611 
.3638 

.9336 
.9325 
.9315 

.3839 
.3872 
.3906 

2.6051 
2.5826 
2.5605 

1.0712 
1.0724 
1.0736 

2.7904 
2.7695 
2.7488 

69°  C 

50' 
40' 

30' 
40' 
50' 

.3665 
.3692 
.3719 

.9304 
.9293 
.9283 

.3939 
.3973 
.4006 

2.5386 
2.5172 
2.4960 

1.0748 

1.0760 

.1.0773 

2.7285 
2.7085 
2.6888 

30' 
20' 
10' 

22°  0' 

10' 
20' 

.3746 
.3773 
.3800 

.9272 
.9261 
.9250 

.4040 
.4074 
.4108 

2.4751 
2.4545 
2.4342 

1.0785 
1.0798 
1.0811 

2.6695 
2.6504 
2.6316 

68°  0' 

50' 
40' 

30' 

.3827 

.9239 

.4142 

2.4142 

1.0824 

2.6131 

30' 

cos  X 

sin  A- 

cot  X 

tan  X 

cosec  X 

sec  X 

X 

NATURAL  FUNCTIONS.     TABLE  III 


339 


X 

Bin  X 

COS  X 

tan  X 

cot  X 

sec  X 

cosec  X 

30' 
40' 
50' 

.3827 
.  3854. 
.3881 

.9239 
.9228 
.9216 

.4142 
.4176 
.4210 

2.4142 
2.3945 
2.3750 

1.0824 
1.0837 
1.0850 

2.6131 
2.5949 
2.5770 

30' 
20' 

10' 

23°  0' 

10' 
20' 

.3907 
.3934 
.3961 

.9205 
.9194 
.9182 

.4245 
.4279 
.4314 

2.3559 
2.3369 
2.3183 

1.0864 
1.0877 
1.0891 

2.5593 
2.5419 
2.5247 

67°  0' 

50' 
40' 

30' 

40' 
50' 

.3987 
.4014 
.4041 

.9171 
.9159 
.9147 

.4348 
.4383 
.4417 

2.2998 

2.2817 

.2.2637 

1.0904 
1.0918 
1,0932 

2.5078 
2.4912 
2.4748 

30' 
20' 
10' 

24°  0' 

10' 
20' 

.4067 
.4094 
.4120 

.9135 
.9124 
.9112 

.4452 

.4487 
.4522 

2.2460 
2.2286 
2.2113 

1.0946 
1.0961 
1.0975 

2.4586 
2.4426 
2.4269 

66°  0' 

50' 
40' 

30' 
40' 
50' 

.4147 
.4173 
.4200 

.9100 
.9088 
.9075 

.4557 
.4592 
.4628 

2.1943 
2.1775 
2.1609 

1.0990 
1.1004 
1.1019 

2.4114 
2.3961 
2.3811 

30' 
20' 
10' 

25°  0' 

10' 
20' 

.4226 
.4253 
.4279 

.9063 
.9051 
.9038 

.4663 
.4699 
.4734 

2.1445 
2.1283 
2.1123 

1.1034 
1.1049 
1.1064 

2.3662 
2.35i5 
2.3371 

65°  0' 

50' 
40' 

30' 
40' 
50' 

.4305 
.4331 
.4358 

.9026 
.9013 
.9001 

.4770 
.4806 
.4841 

2.0965 
2.0809 
2.0655 

1.1079 
1.1095 
1.1110 

2.3228 
2.3088 
2.2949 

30' 
20' 
10' 

26°  0' 

10' 
20' 

.4384 
.4410 
.4436 

.8988 
.8975 
.8962 

.4877 
.4913 
.4950 

2.0503 
2.0353 
2.0204 

1.1126 
1.1142 
1.1158 

2.2812 
2.2677 
2.2543 

64°  0' 

50' 
40' 

30' 
40' 
50' 

.4462 
.4488 
.4514 

.8949 
.8936 
.8923 

.4986 
.5022 
.5059 

2.0057 
1.9912 
1.9768 

1.1174 
1.1190 
1.1207 

2.2412 
2.2282 
2.2154 

30' 
20' 
10' 

27°  0' 

10' 
20' 

.4540 
.4566 
,4592 

.8910 
.8897 
.8884 

.5095 
.5132 
.5169 

1.9626 
1.9486 
1.9347 

1.1223 
1.1240 
1.1257 

2.2027 
2.1902 
2.1779 

63°  0' 

50' 
40' 

30' 
40' 
50' 

.4617 
.4643 
.4669 

.8870 
.8857 
.8843 

.5206 
.5243 
.5280 

1.9210 
1.9074 
1.8940 

1.1274 
1.1291 
1.1308 

2.1657 
2.1537 
2.1418 

30' 
20' 
10' 

28^  0' 

10' 
20' 

.4695 
.4720 
.4746 

.8829 
.8816 
.8802 

.5317 
.5354 
.5392 

1.8807 
1.8676 
1.8546 

1.1326 
1.1343 
1.1361 

2.1301 
2.1185 
2.1070 

62°  0' 

50' 
40' 

30' 
40' 
50' 

.4772 
.4797 
.4823 

.8788 
.8774 
.8760 

.5430 
.5467 
.5505 

1.8418 
1.8291 
1.8165 

1.1379 
1.1397 
1.1415 

2.0957 
2.0846 
2.0736 

.  30' 
20' 
10' 

29°  0' 

10' 
20' 

.4848 
.4874 
.4899 

.8746 
.8732 
.8718 

.5543 
.5581 
.5619 

1.8040 
1.7917 
1.7796 

1.1434 
1.1452 
1.1471 

2.0627 
2.0519 
2.0413 

61°  0' 

50' 
40' 

30' 
40' 
50' 

.4924 
.4950 
.4975 

.8704 
.8689 
.8675 

.5658 
.5696 
.5735 

1.7675 
1.7556 
1.7437 

1.1490 
1.1509 
1.1528 

2.0308 
2.0204 
2.0101 

30' 
20' 
10' 

30°  0' 

.5000 

.8660 

.5774 

1.7321 

1.1547 

2.0000 

60°  0' 

cos  X 

sinX 

cot  X 

tan  X 

cosec  X 

sec  X 

X 

340 


TABLE  III.  NATURAL  FUNCTIONS 


X 

sin  X 

COS  X 

tan  X 

cot  X 

sec  X 

cosec  X 

30°  0' 

10' 
20' 

.5000 
.5025 
.5050 

.8660 
.8646 
.8631 

.5774 
.5812 
.5851 

1.7321 
1.7205 
1.7090 

1.1547 
1.1567 
1.1586 

2.0000 
1.9900 
1.9801 

60°  0' 

50' 
40' 

30' 
40' 
50' 

.5075 
.5100 
.5125 

.8616 
.8601 

.8587 

.5890 
.5930 
.5969 

1.6977 
1.6864 
1.6753 

1.1606 
1.1626 
1.1646 

1.9703 
1.9606 
1.9511 

30' 
20' 
10' 

31°  0' 

10' 
20' 

.5150 
.5175 
.5200 

.8572 
.8557 
.8542 

.6009 
.6048 
.6088 

1.6643 
1.6534 
1.6426 

1.1666 
1.1687 
1.1708 

1.9416 
1.9323 
1.9230 

59°  0' 

50' 
40' 

30'- 
40' 
50' 

.5225 
.5250 
.5275 

.8526 
.8511  , 
.8496 

.6128 
.6168 
.6208 

1.6319 
1.6212 
1.6107 

1.1728 
1.1749 
1.1770 

1.9139 
1.9049 
1.8959 

30' 
20' 
10' 

32°  0' 

10' 
20' 

.5299 
.5324 
.5348 

.8480 
.8465 
.8450 

.6249 
.6289 
.6330 

1.6003 
1.5900 
1.5798 

1.1792 
1.1813 
1.1835 

1.8871 
1.8783 
1 . 8699 

58°  C 

50' 
40' 

30' 
40' 
50' 

.5373 
.5398 

.5422 

.8434 
.8418 
.8403 

.6371 
.6412 
.6453 

1.5697 
1.5597 
1.5497 

1.1857 
1.1879 
1.1901 

1.8612 
1.8527 
1.8444 

30' 
20' 
10' 

33°  0' 

10' 
20' 

.5446 
.5471 
.5495 

.8387 
.8371 
.8355 

.6494 
.6536 
.6577 

1.5399 
1.5301 
1.5204 

1.1924 
1.1946 
1.1969 

1.8361 
1.8279 
1.8198 

57°  0' 

50' 
40' 

30' 
40' 
50' 

.5519 
.5544 
.5568 

.8339 
.8323 
.8307 

.6619 
.6661 
.6703 

1.5108 
1.5013 
1.4919 

1.1992 
1.2015 
1.2039 

1.8118 
1.8039 
1.7960 

30' 
20' 
10' 

34°  0' 

10' 
20' 

.5592 
.5616 
.5640 

.8290 
.8274 
.8258 

.674-5 
.6787 
.6830 

1.4826 
1.4733 
1.4641 

1.2062 
1.2086 
1.2110 

1.7883 
1.7806 
1.7730 

56°  0' 

50' 
40' 

30' 
40' 
50' 

.5664 
.5688 
.5712 

.8241 
.8225 
.8208 

.6873 
.6916 
.6959 

1.4550 
1.4460 
1.4370 

•1.2134 
1.2158 
1.2183 

1.7655 
1.7581 
1.7507 

30' 
20' 
10' 

35°  0' 

10' 
20' 

.5736 
.5760 
.5783 

.8192 
.8175 
.8158 

.7002 
.7046 
.7089 

1-.4281 
1.4193 
1.4106 

1.2208 
1.2233 
1.2258 

1.7435 
1.7362 
1.7291 

55°  0' 

50' 
40' 

30' 
40' 
50' 

.5807 
.5831 
.5854 

.8141 
.8124 
.8107 

.7133 
.7177 
.7221 

1.4019 
1.3934 
1.3848 

1.2283' 

1.2309 

1.2335 

1.7221 
1.7151 
1.7082 

30' 
20' 
10' 

36°  0' 

10' 
20' 

.5878 
.5901 
.5925 

.8090 
.8073 
.8056 

.7265 
.7310 
.7355 

1.3764 
1.3680 
1.3597 

1.2361 
1.2387 
1.2413 

1.7013 
1.6945 
1.6878 

54°  0' 

50' 

40' 

30' 
40' 
50' 

.5948 
.5972 
.5995 

.8039 
.8021 
.8004 

.7400 
.7445 
.7490 

1.3514 
1.3432 
1.3351 

1.2440 
1.2467 
1.2494 

1.6812 
1.6746 
1.6681 

30' 
20' 
10' 

37°  0' 

10' 
20' 

.6018 
.6041 
.6065 

.7986 
.7969 
.7951 

.7536 
.7581 
.7627 

1.3270 
1.3190 
1.3111 

1.2521 
1.2549 
1.2577 

1.6616 
1.6553 
1 . 6489 

53°  0' 

50' 

40' 

30' 

.6088 

.7934 

.7673 

1.3032 

1.2605 

1.6427 

30' 

cos  A- 

sinX 

cot  X 

tanX 

cosec  X 

sec  X 

X 

NATURAL  FUNCTIONS.     TABT.E  III 


341 


X 

1  sin  X 

cos  A- 

tan  X 

cot  X 

sec  X 

coaoc  A 

30' 
40' 
50' 

.6088 
.6111 
.6134 

.7934 
.7916 
.7898 

.7673 
.7720 
.7766 

1.3032 
1.2954 
1.2876 

1.2605 
1.2633 
1.2662 

1.6427 
1.6305 
1.6304 

30' 
20' 
10' 

38°  0' 

10' 
20' 

.6157 
.6180 
.6202 

.7880 
.7862 
.7844 

.7813 
.7860 
.7907 

1.2799 
1.2723 
1.2647 

1.2690 
1.2719 
1.2748 

1.6243 
1.6183 
1.6123 

62°  0' 

50' 
40' 

30' 
40' 
50' 

.6225 
.6248 
.6271 

.7826 
.  7808 
.7790 

.7954 
.8002 
.8050 

1.2572 
1.2497 
1.2423 

1.2779 
1.2808 
1.2837 

1.6064 
1.6005 
1.5948 

30' 
20' 
10' 

39°  0' 

10' 
20' 

.6293 
.6316 
.6338 

.7771 
.7753 
.7735 

.8098 
.8146 
.8195 

1.2349 
1.2276 
1.2203 

1.2868 
1.2898 
1.2929 

1.5890 
1.5833 
1.5777 

151'  0' 
50' 
40' 

30' 
40' 
50' 

.6361 
.6383 
.6406 

.7716 
.7698 
.7679 

.8243 
.8292 
.8342 

1.2131 
1.2059 
1.1988 

1.2960 
1.2991 
1.3022 

1.5721 
1.5666 
1.5611 

30' 
20' 
10' 

40°  0' 

10' 
20' 

.6428 
.6450 
.6472 

.7660 
.7642 
.7623 

.8391 
.8441 
.8491 

1.1918 
1.1847 
1.1778 

1.3054 
1.3086 
1.3118 

1.5557 
1.5504 
1.5450 

50°  0' 

50' 
40' 

30' 
40' 
50' 

.6494 
.6517 
.6539 

.7604 
.7585 
.7566 

.8541 
.8591 
.8642 

1.1708 
1.1640 
1.1571 

1  3151 
1.3184 
1.3217 

1.5398 
1.5346 
1.5294 

30' 
20' 
10' 

41°  0' 

10' 
20' 

.6561 
.6583 
.6604 

.7547 
.7528 
.7509 

.8693 
.8744 
.8796 

1.1504 
1.1436 
1.1369 

1.3250 
1.3284 
1.3318 

1.5243 
1.5192 
1.5142 

49°  0' 

50' 
40' 

30' 
40' 
50' 

.6626 
.6648 
.6670 

.7490 
.7470 
.7451 

.8847 
.8899 
8952 

.9004 
.9057 
.9110 

1.1303 
1.1237 
1.1171 

1.3352 
1.3386 
1.3421 

1.5092 
1.5042 
1.4993 

30' 
20' 
10' 

42°  0' 

10' 
20' 

.6691 
.6713 
.6734 

.7431 
.7412 
.7392 

1.1106 
M041 
1.0977 

1.3456 
1.3492 
1.3527 

1.4945 
1.4897 
1.4849 

48°  0' 

50' 
40' 

30' 
40' 
50' 

.6756 
.6777 
.6799 

.7373 
.7353 
.7333 

.9163 
.9217 
.9271 

1.0913 
1.0850 
1.0786 

1.3563 
1.3600 
1.3636 

1.4802 
1.4755 
1.4709 

30' 
20' 
10' 

43°  0' 

10' 
20' 

.6820 
.6841 
.6862 

\7314 
.7294 
.7274 

.9325 
.9380 
.9435 

1.0724 
1.0661 
1.0599 

1.3673 
1.3711 
1.3748 

1.4663 
1.4617 
1.4572 

47°  0' 

50' 
40' 

30' 
40' 
50' 

.6884 
.6905 
.6926 

.7254 
.7234 
.7214 

.9490 
.9545 
.9601 

1.0538 
1.0477 
1.0416 

1.3786 
1.3824 
1.3863 

1.4527 
1.4483 
1.4439 

30' 
20' 
10' 

44°  0' 

10' 
20' 

.6947 
.6967 
.6988 

.7193 
.7173 
.7153 

.9657 
.9713 
.9770 

1.0355 
1.0295 
1.0235 

1.3902 
1.3941 
1.3980 

1.4396 
1.4352 
1.4310 

46°  0' 

50' 
40' 

30' 
40' 
50' 

.7009 
.7030 
.7050 

.7133 
.7112 
.7092 

.9827 
.9884 
.9942 

1.0176 
1.0117 
1.0058 

1.4020 
1.4001 
1.4101 

1.4267 
1.4225 
1.4184 

30' 
20' 
10' 

45°  0' 

.7071 

.7071 

1.0000 

1.0000 

1.4142 

1,4142 

45°  0' 

cos  X 

sin  a: 

cot  X 

tan  X 

cosec  X 

sec  X 

X 

342    TABLE  IV.    DEGREES  TO  RADIANS  AND  CONVERSELY 


n  degrees 

n  minutes 

n  seconds 

n 

n  radians  into 

a 

into  radians 

into  radians 

into  radians 

degree  measure 

0 

0.00000 

0.00000 

0.00000 

1 

0.01745 

0.00029 

0.00000 

0.00001 

0° 

0' 

02" 

2 

0.03491 

0.00058 

0.00001 

0.00002 

0 

0 

04 

3 

0.05236 

0.00087 

0.00001 

0.00003 

0 

0 

06 

4 

0.06981 

0.00116 

0.00002 

0.00004 

0 

0 

08 

5 

0.08727 

0.00145 

0.00002 

0.00005 

0° 

0' 

10" 

6 

0.10472 

0.00175 

0.00003 

0.00006 

0 

0 

12 

7 

0.12217 

0.00204 

0.00003 

0.00007 

0 

0 

14 

8 

0.13963 

0.00233 

0.00004 

0.00008 

0 

0 

17 

9 

0.15708 

0.00262 

0.00004 

0.00009  • 

0 

0 

19 

10 

0.17453 

0.00291 

0.00005 

11 

0.19199 

0.00320 

0.00005 

0.0001 

0° 

0' 

21" 

12 

0.20944 

0.00349 

0.00006 

0.0002 

0 

0 

41" 

13 

0.22689 

0.00378 

0.00006 

0.0003 

0 

1 

02 

14 

0.24435 

0.00407 

0.00007 

0.0004 

0 

1 

23 

15 

0.26180 

0.00436 

0.00007 

0.0005 

0° 

1' 

43" 

16 

0.27925 

0.00465 

0.00008 

0.0006 

0 

2 

04 

17 

0.29671 

0.00495 

0.00008 

0.0007 

0 

2 

24 

18 

0.31416 

0.00524 

0.00009 

0.0008 

0 

2 

45 

19 

0.33161 

0.00553 

0.00009 

0.0009 

0 

3 

06 

20 

0.34907 

0.00582 

0.00010 

21 

0.36652 

0.00611 

0.00010 

0.001 

0° 

03' 

26" 

22 

0.38397 

0.00640 

0.00011 

0.002 

0 

06 

53 

23 

0.40143 

0.00669 

0.00011 

0.003 

0 

10 

19 

24 

0.41888 

0.00698 

0.00012 

0.004 

0 

13 

45 

25 

0.43633 

0.00727 

0.00012 

0.005 

0° 

17' 

11" 

26 

0.45379 

0.00756 

0.00013 

0.006 

0 

20 

38 

27 

0.47124 

0.00785 

0.00013 

0.007 

0 

24 

04 

28 

0.48869 

0.00814 

0.00014 

0.008 

0 

27 

30 

29 

0.50615 

0.00844 

0,00014 

0.009 

0 

30 

56 

30 

0.52360 

0.00873 

0.00015 

31 

0.54105 

0.00902 

0.00015 

0.01 

0° 

34' 

23" 

32 

0.55851 

0.00931 

0.00016 

0.02 

1 

08 

45 

33 

0.57596 

0.00960 

0.00016 

0.03 

1 

43 

08 

34 

0.59341 

0.00989 

0.00016 

0.04 

2 

17 

31 

35 

0.61087 

0.01018 

0.00017 

0.05 

2° 

51' 

53" 

36 

0.62832 

0.01047 

0.00017 

0.06 

3 

26 

16 

37 

0.64577 

0.01076 

0.00018 

0.07 

4 

00 

39 

38 

0.66323 

0.01105 

0.00018 

0.08 

4 

35 

01 

39 

0.68068 

0.01134 

0.00019 

0.09 

5 

09 

24 

40 

0.69813 

0.01164 

0.00019 

41 

0.71558 

0.01193 

0.00020 

0.1 

5° 

43 

46" 

42 

0.73304 

0.01222 

0.00020 

0.2 

11 

27 

33 

43 

0.75049 

0.01251 

0.00021 

0.3 

17 

11 

19 

44 

0.70794 

0.01280 

0.00021 

0.4 

"^ 

55 

6 

DEGREES  TO  RADIANS  AND  C^ONVERSELY.  TABLE  IV    343 


a 

n  degrees 

n  minutes 

n  seconds 

n  radians  into 

into  radians 

into  radians 

into  radians 

fl 

degree  measure 

45 

0.78540 

0.01309 

0.00022 

0.5 

28°  38' 

52" 

46 

0.80285 

0.01338 

0.00022 

0.6 

34     22 

39 

47 

0.82030 

0.01367 

0.00023 

0.7 

40     06 

25 

48 

0.83776 

0.01396 

0.00023 

0.8 

45     50 

12 

49 

0.85521 

0.01425 

0.00024 

0.9 

51     33 

58 

60 

0.87266 

0.01454 

0.00024 

51 

0.89012 

0.01484 

0.00025 

1.0 

57°   17' 

45" 

52 

0.90757 

0.01513 

0.00025 

2.0 

114     35 

30 

53 

0.92502 

0.01542 

0.00026 

3.0 

171     53 

14 

54 

0.94248 

0.01571 

0.00026 

4.0 

229     10 

59 

55 

0.95993 

0.01600 

0.00027 

5.0 

286°  28' 

44" 

56 

0.97738 

0.01629 

0.00027 

6.0 

343     46 

29 

57 

0.99484 

0.01658 

0.00028 

7.0 

401     04 

14 

58 

1.01229 

0.01687 

0.00028 

8.0 

458     21 

58 

59 

1.02974 

0.01716 

0.00029 

9.0 

515     39 

43 

60 

1.04720 

0.01745 

0.00029 

10.0 

572°  57' 

28" 

TABLE    V.     MATHEMATICAL  CONSTANTS 


7r=  3.14159  26535  89793, 
7r2  =  9.86960  44010  89359 
TpS  =  31.00627  66802  99820. 

^/w=   1.77245  38509  05516. 

,        ..            180° 
1  radian  =  = 


-  =  0.31830  98861  83791. 

TT 

-„  =  0.10132  11836  42338. 

^  =  0.03225  15344  33199. 

0.56418  95835  47756. 


V. 


57°.29577  95131, 


10800' 


648000" 


3437'.74677  07849. 


206264". 80624  70964. 


radian.s. 

1°  =  0.01745  32925  19943. 

(1°)2  =  0.00030  46174  19787. 


radian.s. 

r   =  0.00029     08882 

(l')2=  0.00000     00846     159.50 

(1°)3=  0.00000     53165     76934.  (l')3  =  0.00000     00000     24614, 

1"   =  0.00000     48481     36811 

(1")2  =  0.00000     00000     23504. 

sin  1°  =  0.01745     24064~37284 . 

sin  1'  =  0.00029     08882     04563. 

sin  1"  =  0.00000     48481     36811. 


Naperian  base  =  1  + .--  +  |-:t  + 


2.71828     18284     59045. 


M=  0.43429   44819   03252;   logio  n  =  M  loge  n. 
^  =  2.30258   50929   94046;  loge  «  =  j^  logio  n. 


Badia. 

to 
degree  j 

and  CO. 
versplj 
Math 


I-og^  X, , 


344      TABLE  VI.     VALUES  OF  LOG,  a;, 


AND  e" 


X 

lOgeJV 

eoo 

e-* 

X 

logeA 

eo" 

e-a: 

0.00 

—   00 

1.000 

1.000 

2.50 

0.916 

12.18 

0.082 

0.05 

-2.996 

1.051 

0.951 

2.55 

0.936 

12.81 

0.078 

0.10 

-2.303 

1.105 

0.905 

2.60 

0.956 

13.46 

0.074 

0.15 

-1.897 

1.162 

0.861 

2.65 

0.975 

14.15 

0.071 

0.20 

-1.610 

1.221 

0.819 

2.70 

0.993 

14.88 

0.067 

0.25 

-1.386 

1.284 

0.779 

2.75 

1.012 

15.64 

0  064 

0.30 

-1.204 

1.350 

0.741 

2.80 

1.030 

16.44 

0.061 

0.35 

-1.050 

1.419 

0.705 

2.85 

1.047 

17.29 

0.058 

0.40 

-0.916 

1.492 

0.670 

2.90 

1.065 

18.17 

0.055 

0.45 

-0.799 

1.568 

0.638 

2.95 

1.082 

19.11 

0.052 

0.50 

-0.693 

1.649 

0.607 

3.00 

1.099 

20.09 

0.050 

0.55 

-0.598 

1.733 

0.577 

3.05 

1.115 

21.12 

0.047 

0.60 

-0.511 

1.822 

0.549 

3.10 

1.131 

22.20 

0.045 

0.65 

-0.431 

1.916 

0.522 

3.15 

1.147 

23.34 

0.043 

0.70 

-0.357 

2.014 

0.497 

3.20 

1.163 

24.53 

0.041 

0.75 

-0.288 

2.117 

0.472 

3.25 

1.179 

25.79 

0.039 

0.80 

-0.223 

2.226 

0.449 

3.30 

1.194 

27.11 

0.037 

0.85 

-0.163 

2.340 

0.427 

3.35 

1.209 

28.50 

0.035 

0.90 

-0.105 

2.460 

0.407 

3.40 

1.224 

29.96 

0.033 

0.95 

-0.051 

2.586 

0.387 

3.45 

1.238 

31.50 

0.032 

1.00 

0.000 

2.718 

0.368 

3.50 

1.253 

33.12 

0.030 

1.05 

+  0.049 

2.858 

0.350 

3.55 

1.267 

34.81 

0.029 

1.10 

0.095 

3.004 

0.333 

3.60 

1.281 

36.60 

0.027 

1.15 

0.140 

3.158 

0.317 

3.65 

1.295 

38.47 

0.026 

1.20 

0.182 

3.320 

0.301 

3.70 

1.308 

40.45 

0.025 

1.25 

0.223 

3.490 

0.287 

3.75 

1.322 

42.52 

0.024 

1.30 

0.262 

3.669 

0.273 

3.80 

1.335 

44.70 

0.022 

1.35 

0.300 

3.857 

0.259 

3.85 

1.348 

46.99 

0.021 

1.40 

0.337 

4.055 

0.247 

3.90 

1.361 

49.40 

0.020 

1.45 

0.372 

4.263 

0.235 

3.95 

1.374. 

51.94 

0.019 

1.50 

0.406 

4.482 

0.223 

4.00 

1.386 

54.60 

0.018 

1.55 

0.438 

4.711 

0.212 

4.05 

1.399 

57.40 

0.017 

1.60 

0.470 

4.953 

0.202 

4.10 

1.411 

60.34 

0.017 

1.65 

0.501 

5.207 

0.192 

4.15 

1.423 

63.43 

0.016 

1.70 

0.531 

5.474 

0.183 

4.20 

1.435 

66.69 

0.015 

1.75 

0.560 

5.755 

0.174 

4.25 

1.447 

70.11 

0.014 

1.80 

0.588 

6.050 

0.165 

4.30 

1.459 

73.70 

0.014 

1.85 

0.615 

6.360 

0.157 

4.35 

1.470 

77.48 

0.013 

1.90 

0.642 

6.686 

0.150 

4.40 

1.482 

81.45 

0.012 

1-.95 

0.668 

7.029 

0.142 

4.45 

1.493 

85.63 

0.012 

2.00 

0.693 

7.389 

0.135 

4.50 

1.504 

90.02 

0.011 

2.05 

0.718 

7.768 

0.129 

4.55 

1.515 

94.63 

0.011 

2.10 

0.742 

8.166 

0.122 

4.60 

1.526 

99.48 

0.010 

2.15 

0.766 

8.585 

0.116 

4.65 

1.537 

104.58 

0.010 

2.20 

0.789 

9.025 

0.111 

4.70 

1.548 

109.95 

0.009 

2.25 

0.811 

9.488 

0.105 

4.75 

1.558 

115.58 

0  009 

2.30 

0.833 

9.974 

0.100 

4.80 

1.569 

121.51 

0.008 

2.35 

0.854 

10.486 

0.095 

4.85 

1.579 

127.74 

0.008 

2.40 

0.876 

11.023 

0.091 

4.90 

1.589 

134.29 

0.007 

2.45 

0.896 

11.588 

0.086 

4.95 

1.599 

141.17 

0.007 

2.50 

0.916 

12.182 

0.082 

5.00 

1.609 

148.41 

0.007 

TABLE  VII.  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS  345 


/( 

«2 

Vn 

%Jn 

n2 

n' 

v^ 

^n 

1 

1 

1 

1 

1 

51 

2601 

132651 

7.141 

3.708 

2 

4 

8 

1.414 

1.260 

52 

2'04 

140608 

7.211 

3.733 

3 

9 

27 

1.732 

1.442 

53 

2809 

148877 

7.280 

3.756 

4 

16 

64 

2.000 

1.587 

54 

2916 

157464 

7.348 

3.780 

5 

25 

125 

2.236 

1.710 

55 

3025 

166375 

7.416 

3.803 

6 

36 

216 

2.449 

1.817 

56 

3136 

175616 

7.483 

3.826 

7 

49 

343 

2.646 

1.913 

57 

3249 

185193 

7.550 

3.849 

8 

64 

512 

2.828 

2.000 

58 

3364 

195112 

7.616 

3.871 

9 

81 

729 

3.000 

2.080 

59 

3481 

205379 

7.681 

3.893 

10 

100 

1000 

3.162 

2.154 

60 

3600 

216000 

7.746 

3.915 

11 

121 

1331 

3.317 

2.224 

61 

3721 

226981 

7.810 

3.936 

12 

144 

1728 

3.464 

2.289 

62 

3844 

238328 

7.874 

3.958 

13 

169 

2197 

3.606 

2.351 

63 

3969 

250047 

7.937 

3.979 

14 

196 

2744 

3.742 

2.410 

64 

4096 

262144 

8.000 

4.000 

15 

225 

3375 

3.873 

2.466 

65 

4225 

274625 

8.062 

4.021 

16 

256 

4096 

4.000 

2.520 

66 

4356 

287496 

8.124 

4.041 

17 

289 

4913 

4.123 

2.571 

67 

4489 

300763 

8.185 

4.062 

IS 

324 

5832 

4.243 

2.621 

68 

4624 

314432 

8.246 

4.082 

19 

361 

6859 

4.359 

2.668 

69 

4761 

328509 

8.307 

4.102 

20 

400 

8000 

4.472 

2.714 

70 

4900 

343000 

8.367 

4.121 

21 

441 

9261 

4.583 

2.759 

71 

5041 

357911 

8.426 

4.141 

22 

484 

10648 

4.690 

2.802 

72 

5184 

373248 

8.485 

4.160 

23 

529 

12167 

4.796 

2.844 

73 

5329 

389017 

8.544 

4.179 

24 

576 

13824 

4.899 

2.884 

74 

5476 

405224 

8.602 

4.198 

25 

625 

15625 

5.000 

2.924 

75 

5625 

421875 

8.660 

4.217 

26 

676 

17576 

5.099 

2.962 

76 

5776 

438976 

8.718 

4.236 

27 

729 

19683 

5.196 

3.000 

77 

5929 

456533 

8.775 

4.254 

28 

784 

21952 

5.291 

3.037 

78 

6084 

474552 

8.832 

4.273 

29 

841 

24389 

5.385 

3.072 

79 

6241 

493039 

8.888 

4.291 

30 

900 

27000 

5.477 

3.107 

80 

6400 

512000 

8.944 

4.309 

31 

961 

29791 

5.568 

3.141 

81 

6561 

531441 

9.000 

4.327 

32 

1024 

32768 

5.657 

3.175 

82 

6724 

551368 

9.055 

4.344 

33 

1089 

35937 

5.745 

3.208 

83 

6889 

571787 

9.110 

4.362 

34 

1156 

39304 

5.831 

3.240 

84 

7056 

592704 

9.165 

4.380 

35 

1225 

42875 

5.916 

3.271 

85 

7225 

614125 

9.220 

4.397 

36 

1296 

46656 

6.000 

3.302 

86 

7396 

636056 

9.274 

4.414 

37 

1369 

50653 

6.083 

3.332 

87 

7569 

658503 

9.327 

4.431 

38 

1444 

54872 

6.164 

3.362 

88 

7744 

681472 

9.381 

4.448 

39 

1521 

59319 

6.245 

3.391 

89 

7921 

704969 

9.434 

4.465 

40 

1600 

64000 

6.325 

3.420 

90 

8100 

729000 

9.487 

4.481 

41 

^681 

68921 

6.403 

3.448 

91 

8281 

753571 

9.539 

4.498 

42 

1764 

74088 

6.481 

3.476 

92 

8464 

778688 

9.592 

4.514 

43 

1849 

79507 

6.557 

3.503 

93 

8649 

804357 

9.644 

4.531 

44 

1936 

85184 

6.633 

3.530 

94 

8836 

830584 

9.695 

4.547 

45 

2025 

91125 

6.708 

3.557 

95 

9025 

857375 

9.747 

4.563 

46 

2116 

97336 

6.782 

3.583 

96 

9216 

884736 

9.798 

4.579 

47 

2209 

103823 

6.856 

3.609 

97 

9409 

912673 

9.849 

4.595 

48 

2304 

110592 

6.928 

3.634 

98 

9604 

941192 

9.899 

4.610 

49 

2401 

117649 

7.000 

3.659 

99 

9801 

970299 

9.950 

4.626 

50 

2500 

125000 

7.071 

3.684 

100 

10000 

1000000 

10.000 

4.642 

11 

n2 

n3 

yjn 

^ 

n 

n- 

n3 

v« 

^n 

QyiU^  oru44^i^'f 


^'^.i'-n^-j^i^'m^-'Mi 


5.sM^.''; 


i 


U  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

RENEWALS  ONLY— TEL.  NO.  642-3405 

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General  Library 

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I  illli 

liili 

M 

1       il 

nil 

CDtDl3MfllES 


J 


5C>  V 


Cv 


